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Interaction tests and paired interaction tests in swish
Last seen 8 months ago
United States
I have pairs of rna-seq samples from many cells that were exposed to different targeting and non-targeting knockdowns against three different genes. Each pair consists of material that come from the
inner part of the cell and the outer part of the cell. So we don't expect them pairs to be the same or have the same baseline, but we are interested in the differences between those locations. My
understanding of pairs in fishpond means that I should probably use pairing because it assumes the baselines are the same. Therefore, if we use pairing the differences will be highlighted, which I
I want to run three different tests:
1. targeting vs non-targeting guides for the same gene target
2. inner cellular zone vs the cellular periphery for each knockdown guide
3. A look at the interaction between knockdown guide and cellular zone
After poring over the documentation, I am still unclear whether I should use a covariate term and/or a pairing term for each of these.
I have done the following so far:
1. One simple condition term for test 1
2. One simple condition term for test 2
3. A condition plus pairing for test 3. The documentation suggests that should be equivalent to an interaction term. Is that correct? Because it only allows 1 sample per condition for the pairing, I
would expect that I should actually include the covariate term so that I can do a four-way comparison (2 guides x 2 cellular zones). The covariate term is actually something this model considers
a batch effect and will attempt to correct for, isn't that correct?
Finally, I am a little unclear about which output I should be most concerned with for each question. Currently, I am looking at LFC. But the actual test statistic may be more valuable since it seems
to include both likelihood, such as the q-value, and the the difference, as in the LFC. I would expect the Mann-Whitney Wilcoxin to give a U and a p-value, not a q-value.
Is my current approach the correct one for what I hope to learn?
Entering edit mode
When I attempt to edit my original question, it says there is a field that is required, but I cannot find it, so apologies if this is not the right place to respond.
My experimental design can be summarized with the following table:
**pair** **condition** **cellular zone**
1 treatment soma
1 treatment neurite
2 mock-treatment soma
2 mock-treatment neurite
The soma and neurite samples from each pair come from the same cell. Even though we expect those samples to be different, we are interested in their differences, so the pair term seems appropriate.
I want to run three different tests:
1. treatment vs mock treatment in the same cellular zone
2. neurite vs. soma within the same condition
3. The interaction between condition and cellular zone
How I have approached this so far:
1. swish(y, x="condition") with the input subset to the same cellular zone
2. swish(y, x-"cellular zone") with the input subset to the same condition
3. swish(y, x="condition", pair="pair") with the input subset to treatment_soma vs mock-treatment neurite The documentation states that using the pair term is analagous to creating an interaction
term. So I tried to do that. I don't want to control for a batch effect so I have avoided using the cov term. The documentation also says that using a paired term considers both items in a pair
to have the same baseline, which I think I want because I want to compare the soma vs neurite, and those samples did come from the same cell. However, despite all those reasons for setting up the
test that way, having to artificially subset the samples the way I did seems like it would take away some of the variation I am interested in. So, did I do it correctly?
ADD REPLY • link
Entering edit mode
I think there is a temporary bug in the edit button on the support site, this works. Of your approaches:
1 - looks correct 2 - I would use swish(y, x="cellular_zone", pair="pair") so that you have more power by accounting for the pairs 3 - I would only include pair if you were comparing across cellular
zone (CZ), so not exactly.
If you want to know if if the CZ effect changes across condition that would be:
swish(y, x="cellular_zone", cov="condition", pair="pair", interaction=TRUE)
As in here: https://thelovelab.github.io/fishpond/articles/swish.html#interaction-designs
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SAT Math: An Overview
There are actually two parts on the SAT that include math – a 25-minute non-calculator section and a 55-minute calculator section. Now, within both of these sections there are two problem types, the
regular multiple choice with four answer choices as well as grid-in questions. In the first 25-minute non-calculator portion you will find 15 multiple choice questions and 5 grid-in questions. The
55-minute calculator portion will consist of 30 multiple choice questions and 8 grid-in questions.
Now that you have a general overview of what to expect, let’s talk about what type of math problems to look out for. College Board has broken up the categories into the following: Heart of Algebra,
Problem Solving and Data Analysis, Passport to Advanced Math, and Additional Topics in Math. Heart of Algebra focuses on algebra and linear equations while Problem Solving and Data Analysis measures
your overall math literacy. The Passport to Advanced Math section, on the other hand, focuses on more complex math problems including complex equations. The final topic covered within the math
section is going to be Additional Topics in Math, which addresses both geometric and trigonometric concepts.
Do not get overwhelmed at the range of topics you must master because, in reality, you already know many of these from math class in school. Your first step should be to take a practice test to
diagnose what you need to work on and what you have already mastered. Once you take a practice test, look at the questions you got wrong and try to identify which section that was discussed above it
falls into. Usually, it turns out that they will group together and all be the same underlying concept, and all you have to do is brush up on your volume formulas and suddenly you are getting 3 more
questions correct.
Wondering what type of questions make up the majority of the math sections? Most agree that the highest percentage involves solving single variable equations, with a frequency of roughly 12%, and the
next highest being defining and interpreting linear functions, at a frequency of roughly 11%. The least common skills that appear on the math section include function notation and solid geometry.
As you may be aware, they recently redesigned the SAT and due to this the number of official practice tests available are limited. However, there are still 8 available, and you should utilize each
one to its full potential. Once you click on the following link, scroll down and practice away! The 8 Practice Tests include the questions, answers, as well as explanations to each question.
After taking the practice exams, you can compare your results to the average scores at your preferred universities/colleges. ThoughCo. provides a listing here.
As you go forth in beginning to prepare for the SAT, utilize all of your resources and ensure that you take practice exams under test conditions- including timing yourself. College Board also
provides an app for your phone that allows you to instantly score your practice exams and it even has a “One Question a Day” feature that makes studying for the SAT a little more fun. Check out the
app here! Good luck!
Dhara S. is one of our most experience test prep tutors. Click here to learn more about SAT prep.
Well, it’s true arriving at a useful answer to this question does depend a lot on your timing and the context. If you...
This is a guest post by Christian Heath, author of SAT Math Mastery Volume 1 and Volume 2. Chris has been... | {"url":"https://www.myguruedge.com/our-thinking/actsat-and-applying-to-college/sat-math-an-overview","timestamp":"2024-11-01T20:44:37Z","content_type":"text/html","content_length":"97149","record_id":"<urn:uuid:551462d6-0aeb-41e9-96c2-74bf059732bc>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00320.warc.gz"} |
Converting Tenth of a Second to Other Time Units: Quick Reference
Table of Contents :
Understanding time conversions is crucial for various fields such as sports, science, and everyday time management. One commonly used unit, particularly in contexts like sports timing, is the tenth
of a second. In this guide, we’ll delve into converting tenths of a second into other time units, offering a quick reference and detailed insights.
What is a Tenth of a Second? ⏱️
A tenth of a second (0.1 seconds) is a unit of time that is widely used in many applications. It represents one-tenth of a second and is especially significant in timing sports events where
milliseconds can make a difference in winning or losing.
Why Convert Tenth of a Second? 🤔
• Precision: In competitive environments like athletics, precise timing can distinguish between medals.
• Standardization: Different industries may use various time standards. Converting tenths of a second makes communication clearer.
• Data Analysis: For statistical analysis or performance evaluations, converting time units can provide meaningful insights.
Conversion Table: Tenth of a Second to Other Time Units 📊
Time Unit Conversion Factor Tenth of a Second (0.1 s)
Seconds 1 0.1
Milliseconds 1000 100
Minutes 60 seconds 0.0016667
Hours 3600 seconds 0.00002778
Days 86400 seconds 0.0000011574
Microseconds 1,000,000 100,000
Nanoseconds 1,000,000,000 100,000,000
How to Convert Tenth of a Second to Other Time Units 🔄
Converting to Seconds
To convert tenths of a second to seconds, you simply divide by 10:
[ \text{Seconds} = \frac{\text{Tenths of a Second}}{10} ]
[ 0.1 \text{ s} = 0.1 \div 10 = 0.01 \text{ seconds} ]
Converting to Milliseconds
For converting tenths of a second to milliseconds, you multiply by 1000:
[ \text{Milliseconds} = \text{Tenths of a Second} \times 100 ]
[ 0.1 \text{ s} = 0.1 \times 100 = 100 \text{ milliseconds} ]
Converting to Minutes
To convert tenths of a second to minutes, divide the tenths by 600:
[ \text{Minutes} = \frac{\text{Tenths of a Second}}{600} ]
[ 0.1 \text{ s} = 0.1 \div 600 = 0.00016667 \text{ minutes} ]
Converting to Hours
For hours, divide tenths of a second by 36,000:
[ \text{Hours} = \frac{\text{Tenths of a Second}}{36000} ]
[ 0.1 \text{ s} = 0.1 \div 36000 = 0.00000278 \text{ hours} ]
Converting to Days
Convert to days by dividing by 86,400:
[ \text{Days} = \frac{\text{Tenths of a Second}}{864000} ]
[ 0.1 \text{ s} = 0.1 \div 864000 = 0.00000011574 \text{ days} ]
Important Notes on Time Conversion 📝
Remember: When dealing with time, maintaining accuracy is key. Always double-check your conversions, especially when precision is crucial.
Practical Applications of Tenth of a Second Conversions 🎯
1. Athletics: Track events frequently use tenths of a second for timing sprints and races.
2. Technology: Computing and data processing often require high precision, where timing plays a critical role.
3. Science Experiments: In laboratory settings, measuring time intervals accurately can impact the results.
Tools for Conversion 🛠️
While manual calculations are valuable, several online tools and calculators can assist with quick conversions. These tools can handle conversions automatically and allow for input in various
Summary of Conversion Methods
Below is a quick summary of how to convert tenths of a second to various time units:
Target Unit Conversion Method
Seconds Divide by 10
Milliseconds Multiply by 100
Minutes Divide by 600
Hours Divide by 36,000
Days Divide by 86,400
Microseconds Multiply by 100,000
Nanoseconds Multiply by 100,000,000
Understanding how to convert tenths of a second into other time units is an essential skill that can enhance your efficiency in various activities. Whether you're engaged in sports, technology, or
scientific research, mastering these conversions ensures you can communicate and interpret time accurately. Keep this reference handy for quick access whenever you need to make precise time
conversions! 🌟 | {"url":"https://tek-lin-pop.tekniq.com/projects/converting-tenth-of-a-second-to-other-time-units-quick","timestamp":"2024-11-08T09:13:40Z","content_type":"text/html","content_length":"85486","record_id":"<urn:uuid:bf9d0731-6a52-4ee6-86d6-035feb59473a>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00582.warc.gz"} |
A dynamical system's approach to Schwarzschild null geodesics
The null geodesics of a Schwarzschild black hole are studied from a dynamical system's perspective. Written in terms of Kerr-Schild coordinates, the null geodesic equation takes on the simple form of
a particle moving under the influence of a Newtonian central force with an inverse-cubic potential. We apply a McGehee transformation to these equations, which clearly elucidates the full phase space
of solutions. All the null geodesics belong to one of the four families of invariant manifolds and their limiting cases, further characterized by the angular momentum L of the orbit: for |L| > |L[c]
|, (1) the set that flow outward from the white hole, turn around, and then fall into the black hole, (2) the set that fall inward from past null infinity, turn around outside the black hole to
continue to future null infinity, and for |L| < |L[c]|, (3) the set that flow outward from the white hole and continue to future null infinity and (4) the set that flow inward from past null infinity
and into the black hole. The critical angular momentum L[c] corresponds to the unstable circular orbit at r = 3M, and the homoclinic orbits associated with it. There are two additional critical
points of the flow at the singularity at r = 0. Though the solutions of geodesic motion and Hamiltonian flow we describe here are well known, what we believe is that a novel aspect of this work is
the mapping between the two equivalent descriptions, and the different insights each approach can give to the problem. For example, the McGehee picture points to a particularly interesting limiting
case of the class (1) that move from the white to black hole: in the L → ∞ limit, as described in Schwarzschild coordinates, these geodesics begin at r = 0, flow along t = constant lines, turn around
at r = 2M, and then continue to r = 0. During this motion they circle in azimuth exactly once, and complete the journey in zero affine time.
All Science Journal Classification (ASJC) codes
• Physics and Astronomy (miscellaneous)
Dive into the research topics of 'A dynamical system's approach to Schwarzschild null geodesics'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/a-dynamical-systems-approach-to-schwarzschild-null-geodesics","timestamp":"2024-11-08T05:14:57Z","content_type":"text/html","content_length":"54661","record_id":"<urn:uuid:777dc0d1-a0ad-4c01-a80d-24a751074f9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00148.warc.gz"} |
Multiplying Fractions by Wholes: Lessons and Activites
Because Texas revamped their TEKS and Standards for Math a couple of years ago, I have been given the opportunity to teach Multiplying Fractions by Wholes to my fifth graders. I am slowly but surely
feeling more and more comfortable with this standard.
This year I have finally figured out that there are really two ways that this standard can be taught.
* Finding a Fraction OF a Whole Number, for example: 3/4 of 8
* Repeated addition of a fraction, for example I feed my dog 1/2 cup every day for 5 days. This is 1/2 times 5 or 1/2 + 1/2 + 1/2 + 1/2 + 1/2
Yes, you can do the standard algorithm for both types of problems, but that is not going to teach the students how to make a pictorial model to match both types of situations and the standard states
they should be able to use a model.
So I decided to teach this standard in two parts.
The first part I taught is the Fraction of a Whole number. For this I used a foldable I created to directly instruct students. I used the “I Do”, “We Do” and “You Do” guided instruction. This was
my first time using this foldable in class and it went beautifully! It really helped the students understand how to circle groups and shade them in to find the fraction of a number. The students
even realized, and so did I, that we were making equivalent fractions!
Here is a direct link to my foldable I used: Multiply Fractions by Whole Numbers Interactive Guided Instruction TEKS 5.3I
The next day we used an interactive Google Slides math project. This was something I made and shared with the students. It still focused on the Fraction of a Whole Number. I allowed students to
work on the project in pairs. This was an amazing and rigorous problem solving project! Not only did students work together to problem solve during the slide show, they also problem solved with
technology. They had to use shapes, scribble, type in text boxes and move items around. This activity really drove the concept home!
I absolutely loved this Google assignment and I can’t wait to do another one in the future. Here is a direct link to my store: Multiply a Whole and a Fraction Interactive Google Classroom Activity
TEKS 5.3I
The following day we worked on the other type of problems that they will see with multiplying fractions by wholes. We discussed, drew models and converted fractions from improper fractions to mixed.
We used guided instruction and then students dove into partner work. We used my laminated work pages and fraction cards to come up with the problems to solve. We used dry erase markers which cuts
down on copies. The students seemed okay on this, so we will do another day of work on this topic.
Here is a direct link to the guided instruction activity: Multiply Fractions by Whole Numbers TEKS 5.3I Small Group Partner Game
Next we practiced the algorithm of multiplying, finding an improper and dividing the improper to make a mixed. We used a partner game I created in order to have fun but still practice the problems.
It is called Connect Four and the students were able to practice multiplying fractions while also having good conversations with their partners.
The kids loved Connect Four! Here is a direct link! Connect Four- Partner Game- Multiplying Fractions by Whole Numbers TEKS 5.3I
After today, I think they are ready to move onto, adding and subtracting fractions with unlike denominators. Wish me luck!
Peace, Love, Math! | {"url":"https://themathchick.net/multiplying-fractions-by-wholes-lessons-and-activites/","timestamp":"2024-11-15T04:00:11Z","content_type":"text/html","content_length":"108982","record_id":"<urn:uuid:e19bd101-5ca5-4d02-a636-1919f3c6c6bc>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00838.warc.gz"} |
Three Sum - DeriveIt
This problem asks you to find 3 numbers that sum to the target.
To do this, you can loop over all possible indexes in the array, and check if they add to the target. To avoid duplicate values, you can make sure the indexes you use are in increasing order i < j <
Here's the code:
This solution works, but it's very inefficient. To optimize solutions like this, you should write one of the variables using the other variables. You can subtract a and b from both sides of the
equation, like this:
This equation tells you that the value of c is completely determined by the values of a and b. This is because c = target - a - b. This means you can eliminate the for-loop over $\tt c$, since you
already know what $\tt c$ is equal to. Here's the code for this:
This code is 99% there, but it's slightly broken - it allows elements to be added with themselves. For example, if nums=[1, 2, 10] and target=3, the result would be True (since 1+1+2=4), even though
the answer should really be False. When we look up c in arr we need to make sure c is different from a and b.
To fix this, you can make it so that you only lookup values of c that are to the left of a and b. This makes it so that c is not equal to a or b.
Here's how you code this up:
Time Complexity $O(n^2)$. The loop takes O(n) * O(n) * O(1) = O(n$^2$) time.
Time Complexity $O(n)$, we store a Set with $n$ elements. | {"url":"https://deriveit.org/coding/three-sum-331","timestamp":"2024-11-12T03:45:36Z","content_type":"text/html","content_length":"103683","record_id":"<urn:uuid:017b09b0-7f4d-4abc-8409-a8afd8829423>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00154.warc.gz"} |
[Solved] =+ (b) the markup percentage (rounded to | SolutionInn
=+ (b) the markup percentage (rounded to two decimal places), and (c) the selling price offlat panel
(b) the markup percentage (rounded to two decimal places), and
(c) the selling price offlat panel displays (rounded to nearest whole dollar).
Fantastic news! We've Found the answer you've been seeking! | {"url":"https://www.solutioninn.com/study-help/survey-of-accounting/b-the-markup-percentage-rounded-to-two-decimal-places-2029406","timestamp":"2024-11-11T16:22:58Z","content_type":"text/html","content_length":"72533","record_id":"<urn:uuid:6ee1bb02-7743-410d-bb64-e4f3f7616e00>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00494.warc.gz"} |
The VN test statistic is in the unbiased case $$VN=\frac{\sum_{i=1}^{n-1}(x_i-x_{i+1})^2 \cdot n}{\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2 \cdot (n-1)} $$ It is known that \((VN-\mu)/\sigma\) is
asymptotically standard normal, where \(\mu=\frac{2n}{n-1}\) and \(\sigma^2=4\cdot n^2 \frac{(n-2)}{(n+1)(n-1)^3}\).
The VN test statistic is in the original (biased) case $$VN=\frac{\sum_{i=1}^{n-1}(x_i-x_{i+1})^2}{\sum_{i=1}^{n}\left(x_i-\bar{x}\right)^2}$$ The test statistic \((VN-2)/\sigma\) is asymptotically
standard normal, where \(\sigma^2=\frac{4\cdot(n-2)}{(n+1)(n-1)}\).
Missing values are silently removed. | {"url":"https://www.rdocumentation.org/packages/DescTools/versions/0.99.57/topics/VonNeumannTest","timestamp":"2024-11-13T12:39:04Z","content_type":"text/html","content_length":"95903","record_id":"<urn:uuid:9af60a85-b8d6-4b21-b0c1-7ec6c2a7bc7a>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00586.warc.gz"} |
A generalization of Horner’s rule for polynomial evaluation for ACM National Meeting 1961
ACM National Meeting 1961
Conference paper
A generalization of Horner's rule for polynomial evaluation
Download paper
A general polynomial of n th degree (1) p(x)=a o +a l x+ ... +a n x n may be evaluated for x = α by use of Horner's rule, i.e., by recursively computing (2) b n = a n b j = a j + αb n+1 j = n-1, ...,
0 from which it follows that (4) p(α) = b o . Horner's rule requires n multiplications and n additions to compute p(α). This is generally accepted as the minimum number of such operations to compute
p(α) although no proof exists except for n @@@@ 4. | {"url":"https://research.ibm.com/publications/a-generalization-of-horners-rule-for-polynomial-evaluation","timestamp":"2024-11-08T04:42:44Z","content_type":"text/html","content_length":"72421","record_id":"<urn:uuid:26b006c2-fbbc-45e2-a8fe-7a11cc4a7ed0>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00256.warc.gz"} |
Cardinal number
From Encyclopedia of Mathematics
transfinite number, power in the sense of Cantor, cardinality of a set
That property of the set which is intrinsic to any set Aleph). If
are equivalent. When
One can define for cardinal numbers the operations of addition, multiplication, raising to a power, as well as taking the logarithm and extracting a root. Thus, the cardinal number
Any cardinal number ordinal number of cardinality
(König's theorem). If in (2) one sets
In particular, for any
A cardinal number continuum hypothesis that the classes of strongly- (or weakly-) inaccessible cardinal numbers coincide. The classes of inaccessible cardinal numbers can be further classified (the
so-called Malo scheme), which leads to the definition of hyper-inaccessible cardinal numbers. The assertion that strongly- (or weakly-) inaccessible cardinal numbers exist happens to be independent
of the usual axioms of axiomatic set theory.
A cardinal number
Every cardinal number less than the first uncountable strongly-inaccessible cardinal number is non-measurable (Ulam's theorem), and the first measurable cardinal number is certainly strongly
inaccessible. However, the first measurable cardinal number is considerably larger than the first uncountable strongly-inaccessible cardinal number (Tarski's theorem). It is not known (1987) whether
the hypothesis that measurable cardinal numbers exist contradicts the axioms of set theory.
[1] P.S. [P.S. Aleksandrov] Alexandroff, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2] G. Cantor, , New ideas in mathematics , Handbook Math. Libraries , 6 (1914) pp. 90–184 (In Russian)
[3] F. Hausdorff, "Grundzüge der Mengenlehre" , Leipzig (1914) (Reprinted (incomplete) English translation: Set theory, Chelsea (1978))
[4] K. Kuratowski, A. Mostowski, "Set theory" , North-Holland (1968)
[5] W. Sierpiński, "Cardinal and ordinal numbers" , PWN (1965) (Translated from Polish)
König's theorem stated above is usually called the König–Zermelo theorem.
[a1] T.J. Jech, "Set theory" , Acad. Press (1978) pp. Chapt. 7 (Translated from German)
[a2] A. Levy, "Basic set theory" , Springer (1979)
How to Cite This Entry:
Cardinal number. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cardinal_number&oldid=12770
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article | {"url":"https://encyclopediaofmath.org/index.php?title=Cardinal_number&oldid=12770","timestamp":"2024-11-14T08:39:46Z","content_type":"text/html","content_length":"34238","record_id":"<urn:uuid:5847a08d-ae71-4935-819b-e8ca282539fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00016.warc.gz"} |
non-affine transformations
Support for non-affine transformations would by nifty. I am thinking right now about circle inversions, but this opens up lots of other new opportunities.
You would need an infinite stack to store all the transformations, but this would still be context-free.
If you are willing to draw point-by-point (CIRCLE) then non-affine transformations can be simulated by disabling context-free purity... but thats really really hard.
Re: non-affine transformations
Nice idea.
Technically it should be possible for a minor variant of CF where the allowed transformations were the Möbius transformations of the complex plane. This would have a similar implementation to the
current CF. In CF the transformations are easy to handle since the group of affine transformations is easily parameterised by a small number (6) of real numbers in a matrix, and the composition of
many affine transformations is an affine transformation.
The same goes for Möbius transformations - a group with 4 complex parameters i.e. 8 real parameters (actually only 6 degrees of freedom, so it is a 6-dimensional real Lie group), nicely able to be
represented via matrix algebra. If you also allowed reflection, then you get inversion in a circle, since z/|z|^2 (the inversion in the unit circle) is the conjugate of 1/z, a Möbius transformation.
The other nice thing about Möbius transformations is that they are conformal, and they include any translation, rotation, scaling.
If you wanted to allow Möbius transformations as well as something non-conformal, such as skewing and different x/y scaling, then I think all bets are off. I suspect that the group of these
transformations (i.e. the group that CF would have to keep track of) would be infinite dimensional, and so CF couldn't keep track of them the way it does with affine. Something like you suggest
(descending down the tree each time) would be required, and I think that CF has discarded much of this info as it goes - it could stack up pretty quickly.
Am I right that the reason for asking this is to get CF to inhabit hyperbolic geometry via the Poincaré disk or Poincaré half-plane? That would be remarkably cool.
Implementation? I think I'm dreaming. I couldn't do it - I've trawled through the CF source, and only understand bits, and I've only got one life. If MVJ was sufficiently excited by the idea, he
might be convinced to introduce a CF::Mobius mode into v4.
Cheers, AK
Re: non-affine transformations
I do want to add support for perspective transforms. Once we have more than one type of transformation then adding Möbius transformations would probably be easy. I don't think I even have to write
too much code. The AGG graphics template library has support for bilinear transformations.
Re: non-affine transformations
Yes I remember you saying perspective calculations were one of the things you were considering. I can see that this would be relatively easy - you just start putting things other than [0 0 1] in the
bottom row of the 3x3 matrix. (I think that's right - maybe the rightmost column if you are doing it that way around)
With Möbius transformations, one advantage is that circles stay circles. The problem would be that you can't easily combine them with non-conformal transformations like skewing, etc, without having
to remember and evaluate the whole list of transformations. But this could be global mode-switch for the design OR be rely on an inherited property of shapes so that warnings are thrown when the two
types of transformations are mixed in a given shape's ancestry.
Re: non-affine transformations
I like the idea of the transformation type being an inherited shape property. It's easier to implement than trying to store a global switch in a global variable that needs to be accessed everywhere.
And it would be cool if a design mixed perspective, affine, and Möbius transformations. The root transformation would be a conformal transformation.
Re: non-affine transformations
OK - that would be nice. The neat thing is that similarity transformations (shift, rot, scale, flip) are in both* categories, so the decision as to what type of transformation a shape and its
descendents uses could be deferred until you hit one that forces the decision one way or the other.
* - here I have perspective and affine in the same category, as each can be handled by real 3x3 matrix multiplication.
I was thinking about how I would prove to myself that the two types of transformation are genuinely irreconcilable -- i.e. that there is no simple representation of an arbitrary composition of mobius
and affine transformations. The crunch is that you can come up with a version of the Smale horseshoe map, which basically wrecks any possibility of doing it. | {"url":"https://www.contextfreeart.org/phpbb/viewtopic.php?f=9&t=977&sid=be21e488b0be151eab4b8389dc70be42","timestamp":"2024-11-08T21:17:45Z","content_type":"text/html","content_length":"41891","record_id":"<urn:uuid:7d640a64-58fe-4d54-bb7c-9f401d12be1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00067.warc.gz"} |
ECCC - Reports tagged with worst-case upper bounds
Evgeny Dantsin, Edward Hirsch, Sergei Ivanov, Maxim Vsemirnov
We survey recent algorithms for the propositional
satisfiability problem, in particular algorithms
that have the best current worst-case upper bounds
on their complexity. We also discuss some related
issues: the derandomization of the algorithm of
Paturi, Pudlak, Saks and Zane, the Valiant-Vazirani
Lemma, and random walk ... more >>> | {"url":"https://eccc.weizmann.ac.il/keyword/14746/","timestamp":"2024-11-12T01:04:55Z","content_type":"application/xhtml+xml","content_length":"20277","record_id":"<urn:uuid:a40620df-0332-4daa-ba7e-ea44ebec1196>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00454.warc.gz"} |
Lesson 2 Skills Practice Volume of Cones Answer Key
Understanding Volume
Volume is a unit used to describe how much space a substance or thing occupies. The usual unit of measurement is a cubic unit, such as a cubic foot, cubic meter, or cubic centimeter. By taking the
item's length, breadth, and height measurements and multiplying them all together, the volume of the object may be calculated.
Calculating Volume of Cones
You may use the following formula to determine a cone's volume: V = (1/3)πr²h Where: V = Volume r = Radius of the cone's base h = Height of the cone (pi) = A mathematical constant roughly equal to
Importance of Knowing Volume
The volume of a liquid or gas may be calculated by counting how much room it takes up in a container. Knowing volume is crucial in a number of industries, such as manufacturing, engineering,
research, and architecture, since it enables us to precisely calculate the quantity of materials required for a certain project.
What is the formula to calculate the volume of a cone?
The formula to calculate the volume of a cone is V = (1/3)πr²h, where V is the volume, r is the radius of the cone's base, h is the height of the cone, and (pi) is a mathematical constant roughly
equal to 3.14. | {"url":"https://tutdenver.com/sat/lesson-2-skills-practice-volume-of-cones-answer-key.html","timestamp":"2024-11-08T20:50:05Z","content_type":"text/html","content_length":"21079","record_id":"<urn:uuid:3de35985-1583-4ce5-af49-c2ffd887adea>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00806.warc.gz"} |
In the following operation definitions:
• an A refers to one of the atomic types.
• a C refers to its corresponding non-atomic type. The atomic_address atomic type corresponds to the void* non-atomic type.
• an M refers to type of the other argument for arithmetic operations. For integral atomic types, M is C. For atomic address types, M is std::ptrdiff_t.
• the free functions not ending in _explicit have the semantics of their corresponding _explicit with memory_order arguments of memory_order_seq_cst.
[ Note: Many operations are volatile-qualified. The “volatile as device register” semantics have not changed in the standard. This qualification means that volatility is preserved when applying these
operations to volatile objects. It does not mean that operations on non-volatile objects become volatile. Thus, volatile qualified operations on non-volatile objects may be merged under some
conditions. — end note ]
Effects: leaves the atomic object in an uninitialized state. [ Note: These semantics ensure compatibility with C. — end note ]
constexpr A::A(C desired) noexcept;
Effects: Initializes the object with the value desired. Initialization is not an atomic operation ([intro.multithread]). [ Note: it is possible to have an access to an atomic object A race with its
construction, for example by communicating the address of the just-constructed object A to another thread via memory_order_relaxed operations on a suitable atomic pointer variable, and then
immediately accessing A in the receiving thread. This results in undefined behavior. — end note ]
#define ATOMIC_VAR_INIT(value) see below
The macro expands to a token sequence suitable for constant initialization of an atomic variable of static storage duration of a type that is initialization-compatible with value. [ Note: This
operation may need to initialize locks. — end note ] Concurrent access to the variable being initialized, even via an atomic operation, constitutes a data race. [ Example:
atomic<int> v = ATOMIC_VAR_INIT(5);
— end example ]
bool atomic_is_lock_free(const volatile A *object) noexcept; bool atomic_is_lock_free(const A *object) noexcept; bool A::is_lock_free() const volatile noexcept; bool A::is_lock_free() const noexcept;
Returns: True if the object's operations are lock-free, false otherwise.
void atomic_init(volatile A *object, C desired) noexcept; void atomic_init(A *object, C desired) noexcept;
Effects: Non-atomically initializes *object with value desired. This function shall only be applied to objects that have been default constructed, and then only once. [ Note: These semantics ensure
compatibility with C. — end note ] [ Note: Concurrent access from another thread, even via an atomic operation, constitutes a data race. — end note ]
void atomic_store(volatile A* object, C desired) noexcept; void atomic_store(A* object, C desired) noexcept; void atomic_store_explicit(volatile A *object, C desired, memory_order order) noexcept;
void atomic_store_explicit(A* object, C desired, memory_order order) noexcept; void A::store(C desired, memory_order order = memory_order_seq_cst) volatile noexcept; void A::store(C desired,
memory_order order = memory_order_seq_cst) noexcept;
Requires: The order argument shall not be memory_order_consume, memory_order_acquire, nor memory_order_acq_rel.
Effects: Atomically replaces the value pointed to by object or by this with the value of desired. Memory is affected according to the value of order.
C A::operator=(C desired) volatile noexcept; C A::operator=(C desired) noexcept;
C atomic_load(const volatile A* object) noexcept; C atomic_load(const A* object) noexcept; C atomic_load_explicit(const volatile A* object, memory_order) noexcept; C atomic_load_explicit(const A*
object, memory_order) noexcept; C A::load(memory_order order = memory_order_seq_cst) const volatile noexcept; C A::load(memory_order order = memory_order_seq_cst) const noexcept;
Requires: The order argument shall not be memory_order_release nor memory_order_acq_rel.
Effects: Memory is affected according to the value of order.
Returns: Atomically returns the value pointed to by object or by this.
A::operator C() const volatile noexcept; A::operator C() const noexcept;
Returns: The result of load().
C atomic_exchange(volatile A* object, C desired) noexcept; C atomic_exchange(A* object, C desired) noexcept; C atomic_exchange_explicit(volatile A* object, C desired, memory_order) noexcept; C
atomic_exchange_explicit(A* object, C desired, memory_order) noexcept; C A::exchange(C desired, memory_order order = memory_order_seq_cst) volatile noexcept; C A::exchange(C desired, memory_order
order = memory_order_seq_cst) noexcept;
Effects: Atomically replaces the value pointed to by object or by this with desired. Memory is affected according to the value of order. These operations are atomic read-modify-write operations (
Returns: Atomically returns the value pointed to by object or by this immediately before the effects.
bool atomic_compare_exchange_weak(volatile A* object, C* expected, C desired) noexcept; bool atomic_compare_exchange_weak(A* object, C* expected, C desired) noexcept; bool
atomic_compare_exchange_strong(volatile A* object, C* expected, C desired) noexcept; bool atomic_compare_exchange_strong(A* object, C* expected, C desired) noexcept; bool
atomic_compare_exchange_weak_explicit(volatile A* object, C* expected, C desired, memory_order success, memory_order failure) noexcept; bool atomic_compare_exchange_weak_explicit(A* object, C*
expected, C desired, memory_order success, memory_order failure) noexcept; bool atomic_compare_exchange_strong_explicit(volatile A* object, C* expected, C desired, memory_order success, memory_order
failure) noexcept; bool atomic_compare_exchange_strong_explicit(A* object, C* expected, C desired, memory_order success, memory_order failure) noexcept; bool A::compare_exchange_weak(C& expected, C
desired, memory_order success, memory_order failure) volatile noexcept; bool A::compare_exchange_weak(C& expected, C desired, memory_order success, memory_order failure) noexcept; bool A
::compare_exchange_strong(C& expected, C desired, memory_order success, memory_order failure) volatile noexcept; bool A::compare_exchange_strong(C& expected, C desired, memory_order success,
memory_order failure) noexcept; bool A::compare_exchange_weak(C& expected, C desired, memory_order order = memory_order_seq_cst) volatile noexcept; bool A::compare_exchange_weak(C& expected, C
desired, memory_order order = memory_order_seq_cst) noexcept; bool A::compare_exchange_strong(C& expected, C desired, memory_order order = memory_order_seq_cst) volatile noexcept; bool A
::compare_exchange_strong(C& expected, C desired, memory_order order = memory_order_seq_cst) noexcept;
Requires: The failure argument shall not be memory_order_release nor memory_order_acq_rel. The failure argument shall be no stronger than the success argument.
Effects: Atomically, compares the contents of the memory pointed to by object or by this for equality with that in expected, and if true, replaces the contents of the memory pointed to by object or
by this with that in desired, and if false, updates the contents of the memory in expected with the contents of the memory pointed to by object or by this. Further, if the comparison is true, memory
is affected according to the value of success, and if the comparison is false, memory is affected according to the value of failure. When only one memory_order argument is supplied, the value of
success is order, and the value of failure is order except that a value of memory_order_acq_rel shall be replaced by the value memory_order_acquire and a value of memory_order_release shall be
replaced by the value memory_order_relaxed. If the operation returns true, these operations are atomic read-modify-write operations ([intro.multithread]). Otherwise, these operations are atomic load
Returns: The result of the comparison.
[ Note: For example, the effect of atomic_compare_exchange_strong is
if (memcmp(object, expected, sizeof(*object)) == 0)
memcpy(object, &desired, sizeof(*object));
memcpy(expected, object, sizeof(*object));
— end note ] [ Example: the expected use of the compare-and-exchange operations is as follows. The compare-and-exchange operations will update expected when another iteration of the loop is needed.
expected = current.load();
do {
desired = function(expected);
} while (!current.compare_exchange_weak(expected, desired));
— end example ]
Implementations should ensure that weak compare-and-exchange operations do not consistently return false unless either the atomic object has value different from expected or there are concurrent
modifications to the atomic object.
Remark: A weak compare-and-exchange operation may fail spuriously. That is, even when the contents of memory referred to by expected and object are equal, it may return false and store back to
expected the same memory contents that were originally there. [ Note: This spurious failure enables implementation of compare-and-exchange on a broader class of machines, e.g., load-locked
store-conditional machines. A consequence of spurious failure is that nearly all uses of weak compare-and-exchange will be in a loop.
When a compare-and-exchange is in a loop, the weak version will yield better performance on some platforms. When a weak compare-and-exchange would require a loop and a strong one would not, the
strong one is preferable. — end note ]
[ Note: The memcpy and memcmp semantics of the compare-and-exchange operations may result in failed comparisons for values that compare equal with operator== if the underlying type has padding bits,
trap bits, or alternate representations of the same value. Thus, compare_exchange_strong should be used with extreme care. On the other hand, compare_exchange_weak should converge rapidly. — end note
The following operations perform arithmetic computations. The key, operator, and computation correspondence is:
— Atomic arithmetic computations
Key Op Computation Key Op Computation
add + addition sub - subtraction
or | bitwise inclusive or xor ^ bitwise exclusive or
and & bitwise and
C atomic_fetch_key(volatile A *object, M operand) noexcept; C atomic_fetch_key(A* object, M operand) noexcept; C atomic_fetch_key_explicit(volatile A *object, M operand, memory_order order) noexcept;
C atomic_fetch_key_explicit(A* object, M operand, memory_order order) noexcept; C A::fetch_key(M operand, memory_order order = memory_order_seq_cst) volatile noexcept; C A::fetch_key(M operand,
memory_order order = memory_order_seq_cst) noexcept;
Effects: Atomically replaces the value pointed to by object or by this with the result of the computation applied to the value pointed to by object or by this and the given operand. Memory is
affected according to the value of order. These operations are atomic read-modify-write operations ([intro.multithread]).
Returns: Atomically, the value pointed to by object or by this immediately before the effects.
Remark: For signed integer types, arithmetic is defined to use two's complement representation. There are no undefined results. For address types, the result may be an undefined address, but the
operations otherwise have no undefined behavior.
C A::operator op=(M operand) volatile noexcept; C A::operator op=(M operand) noexcept;
Effects: fetch_key(operand)
Returns: fetch_key(operand) op operand
C A::operator++(int) volatile noexcept; C A::operator++(int) noexcept;
C A::operator--(int) volatile noexcept; C A::operator--(int) noexcept;
C A::operator++() volatile noexcept; C A::operator++() noexcept;
Returns: fetch_add(1) + 1
C A::operator--() volatile noexcept; C A::operator--() noexcept;
Returns: fetch_sub(1) - 1 | {"url":"https://timsong-cpp.github.io/cppwp/n3337/atomics","timestamp":"2024-11-06T17:24:57Z","content_type":"text/html","content_length":"97576","record_id":"<urn:uuid:a8a7d20b-c3a1-424f-8178-9e95b24bceb9>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00223.warc.gz"} |
A measure-theoretic formulation of statistical ensembles (part 2)
This article follows part 1.
In part 2, I will focus on non-thermal ensembles.
Before I proceed, I need to clarify that almost all ensembles that we actually use in physics are thermal ensembles, including the microcanonical ensemble, the canonical ensemble, and the grand
canonical ensemble (the microcanonical ensemble can be considered as a special case of thermal ensemble where $\vec W^\parallel$ is the trivial).
The theory of thermal ensembles is built by letting the system in question be in thermal contact with a bath. Similarly, if we let the system in question be in non-thermal contact with a bath, we can
get the theory of non-thermal ensembles. An example of non-thermal ensembles that is actually used in physics is the isoenthalpic–isobaric ensemble, where we let the system in question be in
non-thermal contact with a pressure bath.
However, we will see that it is harder to measure-theoretically develop the theory of non-thermal ensembles if we continue to use the same method as in the theory of thermal ensembles.
Introducing non-thermal contact with an example
A thermal contact is a contact between thermal system that conducts heat (while exchanging some extensive quantities). A non-thermal contact is a contact between thermal system that does not conduct
heat (while exchanging some extensive quantities). For reversible processes, thermodynamically and mathematically, heat is equivalent to a form of work, where the entropy is the displacement and
where the temperature is the force. However, this is not true for non-reversible processes because of the Clausius theorem. This should have something to do with the fact that entropy is different
from other extensive quantities (as is illustracted in part 1).
First, I may introduce how we may cope with the reversible processes of two subsystems in non-thermal contact in thermodynamics. As an example, consider a tank of monatomic ideal gas separated into
two parts by a thermally non-conductive, massless, incompressible plate in the middle that can move. The two parts can then adiabatically exchange energy ($U$) and volume ($V$) but not number of
particles ($N$). For one of the parts, we have $0=\delta Q=\mathrm dU+p\,\mathrm dV=\mathrm dU+\frac{2U}{3V}\,\mathrm dV,$ which is good and easy to deal with because it is simply a differential
However, this convenience is not possible for non-reversible processes because then we do not have the simple relation $p=2U/3V$. Actually, the pressure is only well-defined for equilibrium states,
and it is impossible to define a pressure that makes sense during the whole non-reversible process, which involves non-equilibrium states. Therefore, although it seems that the “thermally
non-conductive” condition imposes a stronger restriction on what states can the composite system reach without external sources, it actually does not because the energy exchanged by the subsystems
when they exchange volume is actually arbitrary (as long as it does not violate the second law of thermodynamics) if the process is not reversible.
The possible states of the non-thermally composite system then cannot be simply described by a vector subspace of $W^{(1)}\times W^{(2)}$. If we try to use the same approach as constructing the
thermally composite system to construct the non-thermally composite system, the attempt will fail.
Continuing with our example of a tank of gas. Although the pressure is not determined in the non-reversible process, there is one thing that is certain: the pressure on the plate by the gas on one
side is equal to the pressure on the plate by the gas on the other side. This is because the plate must be massless (otherwise its kinetic energy would be an external source of energy; also, remember
that it is incompressible: this means that it cannot be an external source of volume). Therefore, the relation between the volume exchanged and the energy exchanged is determined as long as at least
one side of the plate is undergoing a reversible process because then the reversible side has determined pressure, which determines the pressure of the other side.
This is the key idea of formulating the non-thermal ensembles without formulating the non-thermally composite system. In a thermal or non-thermal ensemble, the composite system consists of two
subsystems, one of which is the system in question, and the other is the bath which we are in control of. We can let the bath have zero relaxation time (the time for it to reach thermal equilibrium)
so that any process of it is reversible. Then, the pressure (or generally, any other intensive quantities that we are in control of times the temperature) is determined (and actually constant), and
we can express the non-conductivity restriction as $\mathrm dU+p\,\mathrm dV=0,$ where $p$ is the pressure, which is a constant. This is a homogeneous linear equation on $\vec W^{\parallel(1)}$
(whose vectors are denoted as $(\mathrm dU,\mathrm dV)$ in our case) which defines a vector subspace of $\vec W^{\parallel(1)}$, which we call $\vec W^{\parallel\parallel(1)}$. The dimension of $\vec
W^{\parallel\parallel(1)}$ is that of $\vec W^{\parallel(1)}$ minus one. The physical meaning of $\vec W^{\parallel\parallel(1)}$ in this example is the hyperplane of fixed enthalpy.
Note that our bath actually has the fixed intensive quantities $i=\left(1/T,p/T\right)\in\vec W^{\parallel(1)\prime}$, we can rewrite the above equation as $\vec W^{\parallel\parallel(1)} =\left\{s_1
\in\vec W^{\parallel(1)}\,\middle|\,i\!\left(s_1\right)=0\right\}.$ $(1)$ Wait! What does $T$ do here? It is supposed to mean the temperature of the bath, but the temperature of the bath is
irrelevant since the contact is non-thermal. Actually, it is. The temperature of the bath serves as an overall constant factor of $i$, which does not affect $\vec W^{\parallel\parallel(1)}$ as long
as it is not zero or infinite. So far, this means that the temperature of the bath is not necessarily fixed, so the actual number of fixed intensive quantities is the dimension of $\vec W^{\parallel
(1)\prime}$ minus one, which is the same as the dimension of $\vec W^{\parallel\parallel(1)}$. Later we will see that anything that is relevant to the temperature of the bath will finally be
irrelevant to our problem. This seems magical, but you will see the sense in that after we introduce another way of developing the non-thermal ensembles (that do not involve baths and non-thermal
contact) later.
We can define a complement of $\vec W^{\parallel\parallel(1)}$ in $\vec W^{\parallel(1)}$ as $\vec W^{\parallel\perp(1)}$. Then, we have $\vec W^{\parallel(1)}=\vec W^{\parallel\parallel(1)}+\vec W^
{\parallel\perp(1)}$. The space $\vec W^{\parallel\perp(1)}$ is a one-dimensional vector space.
For convenience, define $W^{\star\perp(1)}\coloneqq W^{\perp(1)}+\vec W^{\parallel\perp(1)}$. The vector space $\vec W^{\star\perp(1)}$ associated with it is a complement of $\vec W^{\parallel\
parallel(1)}$ in $\vec W^{(1)}$. To make the notation look more consistent, we can use $\vec W^{\star\parallel(1)}$ as an alias of $\vec W^{\parallel\parallel(1)}$. They are the same vector space,
but $\vec W^{\star\parallel(1)}$ emphasizes that it is a subspace of $\vec W^{(1)}$, and $\vec W^{\parallel\parallel(1)}$ emphasizes that it is a subspace of $\vec W^{\parallel(1)}$. Then, we have $W
^{(1)}=W^{\star\perp(1)}+\vec W^{\star\parallel(1)}$. Every point in $W^{(1)}$ can be uniquely written as a sum of a point in $W^{\star\perp(1)}$ and a vector in $\vec W^{\star\parallel(1)}$. We can
describe the decomposition by a projection $\pi^{\star(1)}:W^{(1)}\to W^{\star\perp(1)}$.
We will heavily use the “$\star$” on the superscripts of symbols. Any symbol labeled with “$\star$” is dependent on $i$ (but independent on an overall constant factor on $i$). You can regard those
symbols to have an invisible “$i$” in the subscript so that you can keep in mind that they are dependent on $i$.
Example. Suppose we have a tank of gas with three extensive quantities $U,V,N$. It is in non-thermal contact with a pressure bath with pressure $p$ so that it can exchange $U$ and $V$ with the bath.
Then, the projection $\pi^{\star(1)}$ projects macrostates with the same enthalpy and number of particles into the same point. Because a complement of a vector subspace is not determined, there are
multiple possible ways of constructing the projection. One possible way is $\pi^{\star(1)}\!\left(U,V,N\right)\coloneqq\left(U+pV,0,N\right).$ Here the fixed intensive quantity $p$ is involved. Note
that this projection is still valid for different temperatures of the bath, so an overall constant factor of $i$ does not affect the projection.
Non-thermal contact with a bath
Now, after introducing non-thermal contact with an example, we can now formulate the non-thermal contact with a bath.
Suppose we have a system $\left(\mathcal E^{(1)},\mathcal M^{(1)}\right)$. The main approach is constructing a composite system out of the composite system for the $\vec W^{\parallel(1)}$-ensemble.
The composite system for the $\vec W^{\parallel(1)}$-ensemble was introduced in part 1. We denote the bath that is in contact with our system as $\left(\mathcal E^{(2)},\mathcal M^{(2)}\right)$.
Consider this projection $\pi^\star:W\to W^{\star\perp}$ (where $W^{\star\perp}$ is an affine subspace of $W$ and the range of $\pi^\star$): $\pi^\star\!\left(e_1,e_2\right) \coloneqq\left(\pi^{\star
(1)}\!\left(e_1\right), \rho_{\pi(e_1,e_2)}\!\left(\pi^{\star(1)}\!\left(e_1\right)\right)\right).$ $(2)$ To ensure that it is well-defined, we need to guarantee that $\pi^{\star(1)}\!\left(e_1\
right)\in W^{\parallel(1)}_{\pi(e_1,e_2)}$ for any $e_1,e_2$, and this is true.
The two spaces $W^{\star\perp}$ and $W^{\perp}$ do not have any direct relation. The only relation between them is that the dimension of $W^{\star\perp}$ is one plus the dimension of $W^{\perp}$ (if
they are finite-dimensional).
What is good about the projection $\pi^\star$ is that it satisfies $\vec W^{\star\parallel(1)}=\vec c^{(1)}\!\left(\vec\pi^\star(0)\right)$. This makes our notation consistent if we construct another
composite system out of $\pi^\star$. Now, consider the composite system of $\left(\mathcal E^{(1)},\mathcal M^{(1)}\right)$ and $\left(\mathcal E^{(2)},\mathcal M^{(2)}\right)$ under the projection $
\pi^\star$. In the notation of the spaces and mappings that are involved in the newly constructed composite system, we write “$\star$” in the superscript.
Just like how $\vec W^{\star\parallel(1)}$ is a subspace of $\vec W^{(1)}$, $\vec W^{\star\parallel(2)}$ is also a subspace of $\vec W^{(2)}$. This means that both $\vec\rho^{-1}\circ\vec\rho^\star$
and $\vec\rho\circ\vec\rho^{\star-1}$ are well-defined. The former maps $\vec W^{\star\parallel(1)}$ to another subspace of $\vec W^{(1)}$, and the latter maps $\vec W^{\star\parallel(2)}$ to another
subspace of $\vec W^{(2)}$.
We can regard the construction of the new composite system as replacing the “plate” between the subsystems in the original composite system from a “thermally conductive plate” to a “thermally
non-conductive plate”. Suppose that in the new situation, the intensive quantities “felt” by subsystem 1 is $i^\star\in\vec W^{\star\parallel(1)\prime}$. Then, because the bath is still the same bath
in the two situations, we have $-i^\star\circ\vec\rho^{\star-1}=-i\circ\vec\rho^{-1}.$ Therefore, $i^\star\coloneqq i\circ\vec\rho^{-1}\circ\vec\rho^\star$ $(3)$ would be a good definition of $i^\
star$. However, actually $i^\star$ is trivial: $i^\star=0.$ $(4)$ This is because 2 shows that $\rho\!\left(W^{\star\parallel(1)}_e\right)=W^{\star\parallel(2)}_e$, and thus $\vec\rho^{-1}\!\left(\
vec\rho^\star\!\left(\vec W^{\star\parallel(1)}\right)\right) =\vec W^{\star\parallel(1)},$ which is the kernel of $i$ by definition.
Because $i^\star$ is trivial, it is irrelevant to the temperature of the bath because it is zero no matter what temperature the bath is at.
Example. Suppose a system described by $U_1,V_1,N_1$ is in non-thermal contact with a pressure bath, and they can exchange energy and volume. The projection $\pi$ is $\pi\!\left
(U_1,V_1,N_1,U_2,V_2,N_2\right) =\left(\frac{U_1+U_2}2,\frac{V_1+V_2}2,N_1,\frac{U_1+U_2}2,\frac{V_1+V_2}2,N_2\right).$ Then, the projection $\pi^\star$ can be $\pi^\star\!\left
(U_1,V_1,N_1,U_2,V_2,N_2\right) =\left(U_1+pV_1,0,N_1,U_2-pV_1,V_1+V_2,N_2\right).$ By choosing a different $\pi^{\star(1)}$ or a different $\pi$, we can get a different $\pi^\star$. They physically
mean the same composite system.
The space $W^\perp$ is four-dimensional, and the space $W^{\star\perp}$ is five-dimensional. We can denote the five degrees of freedom as $U,V,H_1,N_1,N_2$, where $U\coloneqq U_1+U_2$ is the total
energy, $V\coloneqq V_1+V_2$ is the total volume, and $H_1\coloneqq U_1+pV_1$ is the enthalpy of subsystem 1. Then, the projection $\pi^\star$ can be written as $\pi^\star\!\left
(U_1,V_1,N_1,U_2,V_2,N_2\right) =\left(H_1,0,N_1,U-H_1,V,N_2\right).$ We can get $W^{\star\parallel}_e$ by finding the inverse of the projection, where $e\coloneqq\left(H_1,0,N_1,U-H_1,V,N_2\right)$:
$W^{\star\parallel}_e\coloneqq\pi^{\star-1}\!\left(e\right) =\left\{\left(H_1-pV_1,V_1,N_1,U-H_1+pV_1,V-V_1,N_2\right)\middle|\,V_1\in\mathbb R\right\}.$ Because it is parameterized by one real
parameter $V_1$, it is a one-dimensional affine subspace of $W$. Projecting it under $c^{(1)}$ and $c^{(2)}$ will respectively give us $W^{\star\parallel(1)}_e$ and $W^{\star\parallel(2)}_e$: $W^{\
star\parallel(1)}_e \coloneqq\left\{\left(H_1-pV_1,V_1,N_1\right)\middle|\,V_1\in\mathbb R\right\},$ $W^{\star\parallel(2)}_e \coloneqq\left\{\left(U-H_1+pV_1,V-V_1,N_2\right)\middle|\,V_1\in\mathbb
The affine isomorphism $\rho^\star_e$ is then naturally $\rho^\star_e\!\left(H_1-pV_1,V_1,N_1\right)=\left(U-H_1+pV_1,V-V_1,N_2\right).$ Its vectoric form is then $\vec\rho^\star\!\left(-p\,\mathrm
dV_1,\mathrm dV_1,0\right) =\left(p\,\mathrm dV_1,-\mathrm dV_1,0\right).$
Our fixed intensive quantities are $i$, which is defined as $i\!\left(\mathrm dU_1,\mathrm dV_1,0\right)=\frac1T\,\mathrm dU_1+\frac pT\,\mathrm dV_1$. We can then get $i^\star$ by $i^\star\coloneqq
i\circ\vec\rho^{-1}\circ\vec\rho^\star =\left(-p\,\mathrm dV_1,\mathrm dV_1,0\right)\mapsto0.$ This is consistent with Equation 4.
Non-thermal ensembles (bath version)
Now, we can define the non-thermal contact with a bath to be the same as the thermal contact with a bath under $\pi^\star$. Utilizing this definition, we can define the composite system for
non-thermal ensembles.
Definition. A composite system for the non-thermal $\vec W^{\parallel(1)}$-ensemble of the system $\left(\mathcal E^{(1)},\mathcal M^{(1)}\right)$ with fixed intensive quantities $i$ is the same as
the composite system for the thermal $\vec W^{\star\parallel(1)}$-ensemble with fixed intensive quantities $i^\star=0$ (given by Equation 4), where $\vec W^{\star\parallel(1)}$ is defined by Equation
This definition looks very neat. Also, just like how we define the domain of fixed intensive quantities of a thermal ensemble, we can define the domain of fixed intensive quantities of a non-thermal
ensemble to consist of those values that make the integral in the definition of the partition function converge.
Because we already derived the formula of the partition function in part 1 that does not involve information about the bath anymore, we can drop the “$(1)$” in the superscripts. The partition
function of the non-thermal ensemble is then $Z^\star\!\left(e,i^\star\right)=\int_{s\in\vec E^{\star\parallel}_e} \Omega\!\left(e+s\right) \mathrm e^{-i^\star\left(s\right)}\,\mathrm d\lambda^{\
parallel}\!\left(s\right),\quad e\in E^{\star\perp},\quad i^\star\in I^\star_e\subseteq\vec W^{\star\parallel\prime}.$ Here, the $i^\star$ is not fixed at the trivial value $0$ (I abused the notation
here) but actually is an independent variable serving as one of the arguments of the partition function that takes values in $I^\star_e$ (which is not the domain of fixed intensive quantities of the
non-thermal ensemble that was mentioned above).
However, the only meaningful information about this non-thermal ensemble is in the behavior of $Z^\star$ at $i^\star=0$ instead of any arbitrary $i^\star\in I^\star_e$, but we do not know whether $0\
in I^\star_e$ or not. This is then a criterion of judge whether $i$ is in the domain of fixed intensive quantities of the non-thermal ensemble or not. To be clear, we define $J\coloneqq\left\{i\in\
vec W^{\parallel\prime}\,\middle|\, \exists e\in E^{\star\perp}:0\in I^\star_{e}\right\}.$ A problem about this formulation is that it is possible to have two $i$’s that share the same thermal
equilibrium state. In that case, the non-thermal ensemble is not defined.
Because $i^\star=0$, the observed extensive quantities in thermal equilibrium are just $\varepsilon^\circ =e+\left.\frac{\partial\ln Z^\star\!\left(e,i^\star\right)}{\partial i^\star}\right|_{i^\star
=0} =e+\frac{\int_{s\in\left(E-e\right)\cap\vec W^{\star\parallel}} s\Omega\!\left(e+s\right)\mathrm d\lambda^{\parallel}\!\left(s\right)} {\int_{s\in\left(E-e\right)\cap\vec W^{\star\parallel}} \
Omega\!\left(e+s\right)\mathrm d\lambda^{\parallel}\!\left(s\right)},$ $(5)$ and the entropy in thermal equilibrium is just $S^\circ=\ln Z^\star\!\left(e,0\right) =\ln\int_{s\in\left(E-e\right)\cap\
vec W^{\star\parallel}} \Omega\!\left(e+s\right)\mathrm d\lambda^{\parallel}\!\left(s\right).$ $(6)$ We can cancel the parameter $e$ by Equation 5 and 6 to get $S^\circ=\ln Z^\star\!\left(\pi^\star\!
\left(\varepsilon^\circ\right),0\right) =\ln\int_{s\in\left(E-\varepsilon^\circ\right)\cap\vec W^{\star\parallel}} \Omega\!\left(\varepsilon^\circ+s\right)\mathrm d\lambda^{\parallel}\!\left(s\
right).$ $(7)$
What is interesting about Equation 7 is that it actually does not guarantee the intensive variables to be defined in $\vec W^\parallel$. Physically this means that the temperature is not necessarily
defined, unlike the case of thermal ensembles (this is because the thermal contact makes the temperature the same as the bath and thus defined). The thing that is guaranteed is that the intensive
variables are defined in $\vec W^{\star\parallel}$ and they must be zero. Therefore, whenever the intensive variables are defined in $\vec W^\parallel$, it must be parallel to $i$ (and remains the
same if we scale $i$ by an arbitrary non-zero factor). Physically, this means that the system must have the same intensive variables as the bath up to different temperatures.
Non-thermal ensembles (non-bath version)
It may seem surprising that we can define non-thermal ensembles without a bath. How is it possible to fix some features about the intensive variables without a bath? The inspiration is looking at
Equation 1. We can make a guess here: if we contract the system along $\vec W^{\star\parallel}$, the contraction satisfy the equal a priori probability principle. We make this guess because of the
following arguments:
• Mathematically, contraction is a legal new system, so it should also satisfy the axioms that we proposed before.
• Physically, because the temperature of the bath is arbitrary, the different accessible macrostates should not be too different because otherwise the temperature would matter (as appears in the
expression of the partition function).
After finding the equilibrium state of the contraction, we can use the contractional pullback to find the equilibrium state of the original system.
If you do it right, you should get the same answer as Equation 7.
The only axiom that we used is the equal a priori probability principle. Then, we formulated three types of ensembles: microcanonical, thermal, and non-thermal. | {"url":"https://ulysseszh.github.io/physics/2023/05/01/measure-ensemble-2.html","timestamp":"2024-11-13T11:01:46Z","content_type":"text/html","content_length":"450785","record_id":"<urn:uuid:be13f5f1-4ccb-4337-a08c-c8dbab42109c>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00326.warc.gz"} |
Powers And Roots Worksheet
Powers and roots at a glance
Powers, or exponents, are numbers written in superscript and denote repeated multiplication. The simplest case is square numbers where a superscript 2, this shows that a number is multiplied by
itself. For cube numbers, with a superscript 3, the multiplication is carried out three times and so on for any integer (whole number) power.
Rooting is the inverse operation. The square root is the inverse of squaring, the cube root is the inverse of cubing, and so on. The root finds the original number that was multiplied.Β
The laws of indices extend these ideas to explain what happens when we multiply, divide or carry out other operations on expressions containing powers, both numeric and algebraic. We can extend these
ideas further to using negative numbers or fractions as powers, and estimating powers and roots using known facts.
Looking forward, students can then progress to additional number worksheets, for example a fractions worksheet or a decimals worksheet.
Β For more teaching and learning support on Number our GCSE maths lessons provide step by step support for all GCSE maths concepts. | {"url":"https://thirdspacelearning.com/secondary-resources/gcse-maths-worksheet-powers-and-roots/","timestamp":"2024-11-07T04:00:35Z","content_type":"text/html","content_length":"161896","record_id":"<urn:uuid:386b59ec-fada-46e8-a5bd-ce8f86a6307a>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00757.warc.gz"} |
Deeper into dataclasses
Usually, examples using the dataclasses module in Python are rather simple in the use of its features. That by itself is completely fine, but sometimes the implementation can be very tedious and
cumbersome. However, the dataclasses module offers ways to be smarter, which are rarely talked about. With this article, I want to change that. Thus, this article doesn't cover topics like when and
how to use them. There is plenty of material on the Internet to learn about that.
The following code represents a 3D point which can be added to another point and multiplied by a number element-wise. (Note: this makes this object more akin to a vector in my view). An extra feature
is that it also supports iteration and unpacking via the __iter__ method, making the point and a number commutative.
# Baseline solution
from dataclasses import astuple, dataclass
class Point:
x: float
y: float
z: float
def __add__(self, other):
x1, y1, z1 = self
x2, y2, z2 = other
return Point(x1+x2, y1+y2, z1+z2)
def __sub__(self, other):
x1, y1, z1 = self
x2, y2, z2 = other
return Point(x1-x2, y1-y2, z1-z2)
def __mul__(self, scalar):
x, y, z = self
return Point(scalar*x, scalar*y, scalar*z)
def __rmul__(self, scalar):
return self.__mul__(scalar)
def __iter__(self):
return iter(astuple(self))
This is a good implementation, but even when using the fact that the point can be unpacked, it is quite tedious and repetitive. Typing x1, y1, z1 = self for every method is less than ideal. Also,
what if we want to also have points in 2D, 4D or 6D? Well, that's straightforward but error-prone. We have to add/delete attributes/fields definitions and all references to them in the relevant
methods. In this particular case, that would be six lines modified. The first part is the easy and the second one (very) annoying. We could do slightly better and only having to care for the first
part if we want to address other dimensions.
Using dataclasses introspection
Let's be slightly smarter and use more of the tools available in the dataclasses module. In particular, the function fields() which exposes the fields defined in a dataclass. Thus, instead of having
to name each coordinate of the point in the different operation methods, we can iterate over them.
# Introspection based solution
from dataclasses import astuple, dataclass, fields
class Point:
x: float
y: float
z: float
def __add__(self, other):
return Point(*(getattr(self, dim.name)+getattr(other, dim.name) for dim in fields(self)))
def __sub__(self, other):
return Point(*(getattr(self, dim.name)-getattr(other, dim.name) for dim in fields(self)))
def __mul__(self, other):
return Point(*(getattr(self, dim.name)*other for dim in fields(self)))
def __rmul__(self, other):
return self.__mul__(other)
def __iter__(self):
return iter(astuple(self))
To understand how it works, I will focus on the __add__ method. There is quite a bit to unpack. At the core is the call to fields() function, which returns a tuple of 3 field objects, a field object
being how a dataclass represents an attribute. You can (and should) go and check it out in a REPL, this can be done on the class itself or an instance of it. Since we have a tuple, we can iterate
over it and access the name of each attribute.
>>> for field in fields(Point):
... print(field.name)
Then, the next step is to use this to get the values associated with each attribute from the instance as follows
>>> p = Point(1,2,3)
>>>list(getattr(p, field.name) for field in fields(p))
Now we can do the operation between the two point instances that are being operated, and we unpack the generator expression into the arguments of the Point initialization
Point(*(getattr(self, dim.name)+getattr(other, dim.name) for dim in fields(self))
The last step is to put this as the return value of the __add__ method and then to implement the same strategy for the other methods. With this, we achieved the first step in removing the annoyance
of having to touch every method if we change the number of dimensions of the point class. As a bonus, we also got to reduce the total amount of lines of code involved slightly.
An alternative solution
I have to agree that using this level of introspection of a dataclass might be a bit cumbersome, particularly by the use of the getattr function. This would be the solution if we weren't implementing
the __iter__ method. But since we're supporting the iterator protocol, we can do something that perhaps is smarter. Thus, instead of having to iterate over the defined fields, we can iterate over the
point object itself!
# Iterator based solution
import operator
from dataclasses import astuple, dataclass, fields
class Point:
x: float
y: float
z: float
def __iter__(self):
return iter(astuple(self))
def __add__(self, other):
return Point(*(operator.add(*pair) for pair in zip(self,other)))
def __sub__(self, other):
return Point(*(operator.sub(*pair) for pair in zip(self,other)))
def __mul__(self, other):
return Point(*(operator.mul(*pair) for pair in zip(self,other)))
def __rmul__(self, other):
return self.__mul__(other)
To highlight the pivotal piece that the __iter__ method plays in this solution, I moved it to the top. For the rest, the code should be pretty much self-explanatory. I'd say that the use of the
functions defined in the operator module also makes the code clearer.
Code quality metrics
Lately, I've also been tangentially interested in code quality metrics. I have a hypothesis regarding the standard metrics to quantify code quality, namely, that they are not well-tailored for
dynamic languages like Python. In particular with features like decorators.
Let's explore some statistics for the three solutions covered before plus a solution without using dataclasses (classic) which is not shown but easy to get from the naive dataclass based solution.
SLOC MI Rank
Classic 24 53.41 A
Baseline 21 54.28 A
Introspection 16 60.22 A
Iterator 17 100 A
Without diving too deep, the maintainability index increases as we move through the different implementations. This result was something that I intuitively expected. What it's shocking is the value
of 100 for the iterator based solution. This warrants a deeper dive into this later on, as it seems unlikely it's a bug in the radon library and right now I'm too ignorant about this topic to be able
to have an idea why this is the case.
Somehow I would have expected a more significant step between the classic and baseline dataclass solutions. But given that the methods we do implement are the same and the ones we didn't have to
write (__init__, __eq__, __repr__) are rather simple it is understandable that they don't differ much.
Using %timeit on my laptop, I ran a quick benchmark of the addition of two points
>>> p1 = Point3D(1,2,3)
>>> p2 = Point3D(4,5,6)
>>> %timeit p1+p2
for the three solutions implemented in this article, with the following results
mean (µs) std
Baseline 33 5.18 µs
Introspection 6.04 450 ns
Iterator 30.4 3.34 µs
We can say that the introspection based solution is considerably more performant than the other two. Would be interesting to understand better where the win (loses) for the introspection (iterator)
based solution come from.
In any case, this example shows something interesting, we can increase the maintainability index while also increasing the performance. This fact is not necessarily a given, as usually performance
comes at the cost of readability.
What about nD points?
But the situation could still be improved. Let's say you want to be able to have 2D and 3D point living along. We could copy and paste the whole definition, give each class a different name, and be
sure to have the correct number of attributes. But that'd be extremely silly and we could (and should) use inheritance (see the previous post which explores abstract base classes and dataclasses) and
reuse all the code for the operations since we just made them independent of the dimension the point lives in. Since we're exploring dataclasses, I'll return to the first solution.
from abc import ABC
from dataclasses import astuple, dataclass, fields
class BasePoint(ABC):
def __add__(self, other):
return self.__class__(*(getattr(self, dim.name)+getattr(other, dim.name) for dim in fields(self)))
def __sub__(self, other):
return self.__class__(*(getattr(self, dim.name)-getattr(other, dim.name) for dim in fields(self)))
def __mul__(self, other):
return self.__class__(*(getattr(self, dim.name)*other for dim in fields(self)))
def __rmul__(self, other):
return self.__mul__(other)
def __iter__(self):
return iter(astuple(self))
class Point2D(BasePoint):
x: float
y: float
class Point3D(BasePoint):
x: float
y: float
z: float
Excellent, we have now points in any dimension we want with little effort!
A factory of points
But is there a way to make even less work than this? The dataclasses module has a nifty function called make_dataclass which, as its name says, makes dataclasses based on its arguments.
We can try and create a point in 1D
>>> Point1D = make_dataclass('Point1D', [('x',float)], bases=(BasePoint,))
>>> Point1D(1)
Compared to defining the class in a normal way this doesn't seem to be a big win. But what if we want to create an exotic point in 5 dimensions? Well, first we create a list of tuples with the field
names and types and then we use make_dataclass with it.
>>> dims = 5
>>> fields_definition = ((f'x{i}', float) for i in range(dims))
>>> Point5D = make_dataclass('Point5D', fields_definition, bases=(BasePoint,))
>>> Point5D(*range(5))
Point5D(x0=0, x1=1, x2=2, x3=3, x4=4)
I move from the naming xyz to x{i} in a more "mathematical" notation which for computers also works much better.
This sets the stage up to create a whole family of points. For this we create a function which will create them (a Factory)
def PointFactory(dim):
fields_definition = ((f'x{i}', float) for i in range(dim))
return make_dataclass(f'Point{dims}D', fields_definition, bases=(BasePoint,))
Making use of this factory a series of classes representing points in different dimensions can be easily created
>>> point_classes = [PointFactory(dim) for dim in range(5)]
>>> point_classes
[<class 'abc.Point0D'>, <class 'abc.Point1D'>, <class 'abc.Point2D'>, <class 'abc.Point3D'>, <class 'abc.Point4D'>]
>>> point_classes[3](1,2,3)
Point3D(x0=1, x1=2, x2=3)
Besides the boilerplate reduction that the dataclasses module provides, it offers some powerful tooling to work with them. As an example, I showed how to create a factory of n-dimensional points.
Moreover, I discovered that using the introspection machinery of dataclasses leads to a higher performant code. Not that this was a goal of this article, but it's always nice to get a boost. Keep in
mind that introspection, in this case, might be a slightly wrong term, as it would make people believe it should be less performant, particularly those coming from Go.
After seeing the performance of each implementation, the question arises if the iterator based could be improved by being smarter. At least my first exploratory attempt by moving from import operator
to from operator import add, mul, sub did not show any change. Perhaps this would be a good exercise for the reader ;)
I want to thank Nour Faroua for her contribution leading to simplification on the code, and to Bryan Reynaert for his thorough review and input improving organization, explanations, and language of
the article.
Top comments (0)
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Capacity Of Belt Conveyor
Conveyor Capacity - Engineering ToolBox
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Troughing Belt Conveyor Capacity
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Capacity Of Belt Conveyor - two-do.nl
Capacity Of Belt Conveyor. 5 1 introduction this final year project was carried out in a chinese company, chaohu machinery manufacturing cotd and the aim of this thesis was to point out that a better
method to optimizing belt conveyor manufacturing process should be utilized by this.
How to Increase Conveyor Capacity | E & MJ
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Conveyor Belt Calculations Brighthub Engineering. An Example of Conveyor Belt Calculations Input data Conveyor capacity Cc 1500 th 41667 Kgsec Belt speed V 15 msec Conveyor height H 20 m Conveyor
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calculations for capacity of belt conveyor
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how to calculate the capacity of the belt conveyor
Belt Capacity Chart - Supplier & Distributor of … Belt Capacity Chart The Following conveyor belt capacity charts show tons per hour (TPH) based on material weighing 100 lbs. per cubic foot, 20
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An Unusual Periodic Table
Lemniscate The word lemniscate comes from the Greek word lemniskus for ribbon. The mathematical curve, a sort of figure eight, does look somewhat like the bow for a package made from a twisted ribbon
[see figure]. The word is beginning to disappear from textbooks, and is completely missing in my high school edition of the American Heritage Dictionary. The only closely related term I could find
was lemniscus, a term for a nerve bundle in the brain. No picture was available, but it may be this also looks like a ribbon.
The Mathematical curve [formulas below]is related to the rectangular hyperbola through the following relationship. If a tangent is drawn to the hyperbola and the perpendicular to the tangent is drawn
through the origin, the point where the perpendicular meets the tangent is on the lemniscate.
I recently saw a picture of a chemical periodic table in the shape of a lemniscate created by William Crookes in 1888. The picture is on page 107 of The Ingredients: A Guided Tour of the Elements by
Philip Ball. | {"url":"https://pballew.blogspot.com/2020/05/an-unusual-periodic-table.html","timestamp":"2024-11-08T21:59:37Z","content_type":"application/xhtml+xml","content_length":"128129","record_id":"<urn:uuid:48fe8a85-5221-47e2-8f38-78661ae56293>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00607.warc.gz"} |
How to Pick Stocks Using the Discounted Cash Flow (DCF) Method?
Picking stocks using the discounted cash flow (DCF) method is a popular approach for investors who value stocks based on their expected future cash flows. Here is a brief overview of how to use the
DCF method to pick stocks:
1. Understand the concept: The DCF method involves estimating the present value of a stock's future cash flows. Investors believe that the intrinsic value of a stock is the sum of its discounted
future cash flows.
2. Estimate future cash flows: Start by estimating the future cash flows the company is expected to generate. This requires analyzing the company's financial statements, industry trends, growth
prospects, competitive position, and other relevant factors.
3. Determine the appropriate discount rate: The discount rate reflects the opportunity cost of investing in the stock. It represents the rate of return required by an investor for taking on the risk
of investing in the stock market. A higher discount rate reflects higher risk and vice versa.
4. Calculate the present value: Once future cash flows and the discount rate are determined, use a formula to calculate the present value of each cash flow. The formula discounts the future cash
flows by the discount rate to reflect the time value of money.
5. Sum the present values: Add up the present values of all estimated future cash flows to obtain the total intrinsic value of the stock.
6. Compare intrinsic value and market price: Compare the intrinsic value obtained from the DCF analysis with the market price of the stock. If the intrinsic value is higher than the market price,
the stock is considered undervalued and could be a potential investment opportunity. If the intrinsic value is lower than the market price, the stock may be overvalued.
7. Consider other factors: While the DCF method provides a fundamental analysis, it is essential to consider other factors such as qualitative aspects, industry trends, management quality,
competitive advantages, and market sentiment before making a final investment decision.
It is important to note that the DCF method has limitations and is based on assumptions that may not always hold true. Therefore, it's advisable to use the DCF method as one of the tools in your
investment analysis toolbox rather than relying solely on it.
How to analyze stocks using the discounted cash flow (DCF) method?
To analyze stocks using the discounted cash flow (DCF) method, follow these steps:
1. Gather Financial Data: Collect the necessary financial data of the stock you want to analyze, including historical and projected financial statements, such as income statement, balance sheet, and
cash flow statement.
2. Determine Cash Flow: Identify and estimate the company's annual free cash flows. Free cash flow is the money left after deducting operating expenses, taxes, and capital expenditure from revenue.
Use a combination of historical data and future projections to arrive at the estimated free cash flow figures.
3. Set a Discount Rate: Determine the appropriate discount rate to apply to the future cash flows. This rate represents the required rate of return for the investor and should reflect the risk
associated with the investment. A common approach is to use the weighted average cost of capital (WACC), considering the company's cost of debt and equity.
4. Forecast Cash Flows: Project the company's cash flows into the future for a specific period, usually 5-10 years. Ensure that these projections are realistic and consider various factors like
industry trends, competition, market conditions, and potential risks.
5. Calculate Terminal Value: Determine the terminal value, which represents the company's value beyond the projected period. This can be calculated using a terminal growth rate, which assumes a
stable growth rate for the company in perpetuity. The terminal value is calculated based on the free cash flow projected for the last year of the projection period.
6. Discount Cash Flows: Using the discount rate set in step 3, discount each year's projected cash flow and the terminal value back to their respective present values. This involves dividing each
cash flow by the appropriate discount rate for that year.
7. Sum the Present Values: Add up all the present values of the projected cash flows and the terminal value to determine the total value of the stock.
8. Compare to Market Price: Compare the calculated value obtained through the DCF analysis to the current market price of the stock. If the DCF value is higher than the market price, the stock may
be undervalued and potentially a good investment opportunity. Conversely, if the DCF value is lower than the market price, the stock may be overvalued.
It's important to note that DCF analysis relies heavily on the accuracy of the projections and the discount rate used, so careful consideration and due diligence are necessary when conducting this
How to estimate the terminal value in the DCF method?
Estimating the terminal value in the DCF (Discounted Cash Flow) method involves predicting the value of a business beyond the forecasted period. To estimate the terminal value, you can follow these
1. Choose a terminal year: Select a year beyond the forecasted period, usually 5 or 10 years ahead. The choice should align with the industry and business characteristics.
2. Select a suitable growth rate: Determine a sustainable growth rate that the business can achieve in the long term. This growth rate should represent the growth potential of the industry and
company and consider factors like market conditions, competition, and economic outlook.
3. Calculate free cash flow: Determine the forecasted free cash flow for the terminal year. This can be done by extrapolating the cash flow projection from the last forecasted year using the
selected growth rate.
4. Apply a perpetuity formula: Apply the perpetuity formula, also known as the Gordon Growth Model, to calculate the terminal value. The formula is: Terminal Value = FCFT / (Discount Rate - Growth
Rate). FCFT represents the free cash flow in the terminal year, and the discount rate is the same rate used to discount cash flows in the forecasted period.
5. Discount the terminal value: To bring the terminal value to present value, discount it back to the current year using the appropriate discount rate. This discounted terminal value is added to the
present value of the forecasted cash flows to determine the total enterprise value.
Remember that estimating the terminal value involves uncertainties, and small changes in the growth rate or discount rate can significantly impact the valuation outcome. It's crucial to exercise
judgment and consider different scenarios and sensitivities when estimating the terminal value in the DCF method.
What are the key industry-related factors to consider in the DCF analysis?
When performing a discounted cash flow (DCF) analysis, there are several key industry-related factors to consider:
1. Market growth rate: The growth rate of the industry is a crucial factor as it directly impacts the future cash flows of the company. Understanding the industry's growth potential helps in
estimating the company's future revenue and cash flow projections.
2. Competitive landscape: Evaluating the competitive landscape is critical to assess the company's market positioning and its ability to maintain or increase market share. Factors such as market
share concentration, entry barriers, and competitive advantages need to be considered.
3. Industry trends and dynamics: Analyzing industry trends, market cycles, and technological advancements is important to determine the long-term viability of the industry and its impact on the
company's future cash flows.
4. Regulatory environment: Assessing the regulatory framework and understanding any existing or potential regulations that may affect the industry is crucial. Industry-specific regulations can
impact pricing, entry barriers, and cost structures, affecting the company's profitability.
5. Customer preferences: Understanding customer behavior, preferences, and demands within the industry is essential. This knowledge helps in assessing the company's ability to meet customer needs
and adjust its offerings accordingly.
6. Supply chain considerations: Evaluating the industry's supply chain, including suppliers and distribution channels, is important to understand potential risks and impact on cost structures.
Supply chain disruptions can affect the company's cash flow generation.
7. Industry-specific risks: Identifying and evaluating industry-specific risks such as commodity price volatility, technological obsolescence, or changing consumer tastes should be considered. These
risks can affect the company's future cash flows and its overall risk profile.
8. Macroeconomic factors: Analyzing macroeconomic factors like interest rates, inflation rates, and overall economic conditions is important as they can impact the industry's growth rate and the
company's cost of capital used in the DCF model.
Considering these key industry-related factors in a DCF analysis helps in determining the company's future cash flow projections, discount rate, and overall valuation.
What are the key components of the DCF method?
The key components of the DCF (Discounted Cash Flow) method are as follows:
1. Cash Flows: Future cash flows generated by the business or investment are estimated. These cash flows include project revenues, expenses, and any other relevant income or costs.
2. Discount Rate: A discount rate is applied to the future cash flows to account for the time value of money. This rate reflects the riskiness of the investment and represents the minimum return an
investor would require.
3. Terminal Value: The DCF method assumes that cash flows will continue indefinitely, but it is not feasible to project them forever. Therefore, a terminal value is calculated, which represents the
estimated value of the investment at the end of the projected period.
4. Present Value: The future cash flows and terminal value are discounted back to their present value using the discount rate. This calculation adjusts for the fact that money received in the future
is worth less than the same amount received today.
5. Net Present Value: The present value of future cash flows and terminal value is subtracted from the initial investment or cost of the investment. The result is the net present value (NPV), which
indicates whether the investment is expected to generate positive or negative value.
6. Sensitivity Analysis: Sensitivity analysis is performed to assess the impact of different variables and assumptions on the NPV. By testing various scenarios, sensitivity analysis helps identify
the key drivers of value and the impact of changes in assumptions on the investment's viability.
7. Decision Rule: Based on the calculated NPV and sensitivity analysis, a decision is made on whether to proceed with the investment or project. If the NPV is positive and the investment meets the
required return threshold, it is considered a worthwhile investment. | {"url":"https://topminisite.com/blog/how-to-pick-stocks-using-the-discounted-cash-flow","timestamp":"2024-11-04T05:21:12Z","content_type":"text/html","content_length":"276800","record_id":"<urn:uuid:0a16de9d-034b-4146-bb1c-cb83456ce835>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00045.warc.gz"} |
How do I use the graph of a linear function to find its equation? | Socratic
How do I use the graph of a linear function to find its equation?
1 Answer
If you have two points on the graph, you can calculate the equation.
Select any two point on the graph. It may make life easier to have two points that are on gridlines in your graph. If one of the points is the $y$-intercept, it makes life a lot easier.
(the $y$-intercept is the point where your graph crosses the vertical, or $y$-axis)
You want to find the equation $y = m \cdot x + b$
where $m$ is called the slope, and $b$ the $y$-value at the $y$-intercept.
To calculate the slope $m$
Let's call the two points you selected $\left({x}_{1} , {y}_{1}\right)$ and $\left({x}_{2} , {y}_{2}\right)$
Then the slope is how fast $y$ changes relative to a change in $x$
Or in formula (in formulae, $\Delta$ means "change in..."):
$m = \frac{\Delta y}{\Delta x} = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
To calculate the intercept $b$
If one of the points you selected is on the $y$-axis, then this $y$-value is equal to $b$.
Otherwise you fill in the equation $y = m x + b$ with the values for one of the points:
${y}_{1} = m {x}_{1} + b \to b = {y}_{1} - m {x}_{1} = {y}_{1} - \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} {x}_{1}$
(substituting the $m$ you already calculated in the previous step)
One example
Step 1: finding $m$
Your points are $\left(- 4 , - 2\right)$ and $\left(4 , 0\right)$
$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{0 - - 2}{4 - - 4} = \frac{2}{8} = \frac{1}{4}$
Step 2 : finding $b$
Substitute $x$ and $y$ for the right point (because it's easier)
$y = m x + b \to 0 = \frac{1}{4} \cdot 4 + b \to 0 = 1 + b \to b = - 1$
The complete equation will be: $y = \frac{1}{4} x - 1$
Step 3: check! your answer:
Fill in the $x$ of you other point (you should get the proper $y$)
$\frac{1}{4} \cdot \left(- 4\right) - 1 = - 1 - 1 = - 2$
Impact of this question
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Consider a market supply and demand represented by the follo
Consider a market supply and demand represented by the following: [Math Processing Error]Qs=4P−120 and [Math Processing Error]Qd=1000−10P
Subject:MarketingPrice:2.88 Bought3
Consider a market supply and demand represented by the following: [Math Processing Error]Qs=4P−120 and [Math Processing Error]Qd=1000−10P. Use this information to answer the following questions.
a. Calculate the equilibrium price and quantity.
b. What is the consumer surplus?
c. If the government imposes an excise tax of $2, what would be the new equilibrium price and quantity?
d. What would happen to the consumer surplus?
Option 1
Low Cost Option
Download this past answer in few clicks
2.88 USD
Option 2
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TEACH Grant
In this chapter, we will illustrate the amounts a student may receive under the TEACH Grant program and show how to determine the correct grant award for each payment period. For more detail on TEACH
Grant criteria and eligibility, see Volume 1, Student Eligibility. For more on payment periods, see Chapter 1 of this volume, and for cost of attendance, see Chapter 2 of this volume.
• Calculating a TEACH Grant for a payment period
• Calculating TEACH for a payment period that occurs in two award years
• TEACH Grants for transfer students
• Correspondence study and TEACH
• Recalculation of TEACH Grants
The TEACH Grant program is a non-need-based grant program that provides up to $4,000 per year^1 to students who are enrolled in an eligible program and who agree to teach in a high-need field, at a
low- income elementary school, secondary school, or educational service agency^2 as a highly qualified teacher, for at least four years within eight years of completing the program for which the
TEACH Grant is awarded. The student must sign a service agreement to this effect and complete initial counseling prior to receiving a first TEACH Grant and subsequent counseling before receiving each
subsequent TEACH Grant. If the student subsequently fails to meet the requirements of the service agreement, the TEACH Grants will be converted to a Direct Unsubsidized loan that the student must
repay in full, with interest charged from the date of each TEACH Grant disbursement. For more details on the TEACH Grant service agreement, eligibility, and conversion from a grant to a loan, see
Volume 1.
With respect to enrollment status, the program must require an undergraduate student to enroll for at least 12 credit-hours in each term in the award year to qualify as full-time. For a graduate
student, each term in the award year must meet the minimum full-time enrollment status established by your school for a semester, trimester, or quarter.
TEACH Grant resources COD School Relations Center:
^1See the subsection “the Sequester and TEACH Grants” later in this chapter for reductions to the maximum award amount due to the Sequester.
TEACH Grant Scheduled, Annual, and Aggregate Awards
The TEACH Grant award amounts are similar to Pell awards in that there is a Scheduled Award, which is the maximum that a full-time student may receive for a year, and an Annual Award, which is the
amount a student may receive for a year based on enrollment status (i.e., full-time, three- quarter-time, half-time, or less-than-half-time). The Scheduled Award for TEACH is $4,000, and the annual
awards are:
Scheduled Award for TEACH is $4,000, and the annual awards
Full-time .......................................... $4,000
3/4-time........................................... $3,000
1/2-time............................................ $2,000
less-than-1/2-time ...........................$1,000
A student may receive up to $16,000 in TEACH Grants for undergraduate and post-baccalaureate study, and up to $8,000 for a TEACH Grant-eligible master’s degree program.
Calculating Teach Grant Payments For Payment Periods
As for other FSA programs, for purposes of calculating a TEACH Grant for a payment period, the definition of an academic year must include, for undergraduate programs of study (including those
post-baccalaureate programs that are TEACH Grant eligible), both the required credit or clock- hours and weeks of instructional time (see Chapter 1).
The formula you will use to calculate the amount of a student’s TEACH Grant that will be awarded for a payment period depends on the academic calendar used by the student’s program. These formulas
are the same as for Pell Grants, with the exception of master’s degree programs. For details on these payment formulas, see Chapter 3 of this volume. For master’s degree programs, a TEACH Grant
eligible program’s academic year must be defined as at least the required number of weeks of instructional time and the minimum number of credit or clock-hours that a full-time student would be
expected to complete in the weeks of instructional time. Note that no payment for a payment period may be less than $25.
Calculation of a TEACH Grant for a payment period
Crossover payment periods
If a student enrolls in a payment period that is scheduled to occur in two award years, the entire payment period must be considered to occur within one of those award years, and the school must
report TEACH Grant payments to the student for that payment period as being made for the award year to which the crossover payment period was assigned. There is no requirement for a TEACH Grant
crossover payment period to be placed in the same award year as Pell.
In most cases, it is up to the school to determine the award year in which the payment period will be placed. However, if more than six months of a payment period are scheduled to occur within one
award year, you must place that payment period in that award year.
Crossover payment periods
Payment for a payment period from two Scheduled Awards
When a student’s payment period spans two different Scheduled Awards, the student’s payment for the payment period is calculated based on the total credit or clock-hours and weeks of instructional
time in the payment period, and is the remaining amount of the Scheduled Award being completed plus an amount from the next Scheduled Award, (if available) up to the payment for the payment period.
For more details, see 34 CFR 686.22(i).
Payment within payment period & retroactive payment
Within each payment period, you may pay the student at such times and in such installments as you determine will best meet the student’s needs. You may pay a student TEACH Grant funds in one lump sum
for all prior payment periods for which the student was eligible within the award year, as long as the student has signed the agreement to serve prior to disbursement of the TEACH Grant (for more
details on the agreement to serve and TEACH Grant eligibility, see Volume 1).
A student who receives a TEACH Grant at one institution and subsequently enrolls at a second institution may receive a TEACH Grant at the second institution if the second institution obtains the
student’s valid SAR or ISIR with an official EFC.
The second institution may pay a TEACH Grant only for that period in which a student is enrolled in a TEACH Grant-eligible program at that institution. The second institution must calculate the
student’s award using the appropriate formula, unless the remaining balance of the Scheduled Award at the second institution is the balance of the student’s last Scheduled Award and is less than the
amount the student would normally receive for that payment period.
A transfer student must repay any amount received in an award year that exceeds the amount which he or she was eligible to receive. A student may not receive TEACH Grant payments concurrently from
more than one school.
The sequester and TEACH Grants
On August 2, 2011, Congress passed the Budget Control Act (BCA) of 2011, which put into place a federal budget cut known as the sequester. All TEACH awards first disbursed during the federal fiscal
year (FY) 2020 (on or after October 1, 2019, and before October 1, 2020) must be reduced by 5.9% from the award amount the student would otherwise be eligible to receive. In FY 21 (beginning on
October 1, 2020), the reduction is 5.7%. For more details on the sequester and TEACH Grants, see the June 23, 2020 Electronic Announcement.
The requirements for calculating a TEACH Grant payment for a payment period are exactly the same as Federal Pell Grant program requirements and use the same formulas as the Pell Grant program. TEACH
Grant formulas 1, 2, 3, 4, and 5 are identical to the corresponding Pell formulas. The school disburses a TEACH Grant, like Pell, over the hours and weeks of instruction in an eligible program’s
academic year, as defined by the school.
As with Pell Grants, TEACH Grant Scheduled Awards are divided into at least two payments based on the payment periods in a year. The calculation formula you use depends on the academic calendar of a
student’s eligible program and would be the same formula used to calculate payments of Pell Grants for that academic program. For students ineligible for Pell Grants, such as master’s degree
students, you must use the calculation formula that corresponds to the academic calendar of the eligible student’s program. Refer to Chapter 3 of this volume on Pell Grants for a more detailed
explanation of these formulas.
A student’s payment for a payment period is calculated based on the coursework in the student’s TEACH Grant-eligible program. For a TEACH Grant, the school must ensure that the student’s courses are
necessary for the student to complete the student’s TEACH Grant-eligible program.
TEACH Grant formulas
Formula 1: 34 CFR 686.22(a)(1),(b)
Formula 2: 34 CFR 686.22(a)(2),(c)
Formula 3: 34 CFR 686.22(a)(3),(d)
Formula 4: 34 CFR 686.22(a)(4),(e)
Formula 5: 34 CFR 686.25
A student must complete initial counseling before receiving his or her first TEACH Grant and must complete subsequent counseling before receiving each subsequent TEACH Grant. Initial and subsequent
TEACH Grant counseling must be completed on the Department’s Student.Aid.gov website. You will receive reports from the Department on all students who have completed counseling.
You must ensure that TEACH Grant exit counseling is conducted with each TEACH Grant recipient before the student ceases to attend your school at a time that you determine. The exit counseling must be
in person, by audio-visual presentation, or by interactive electronic means (such as on the Department’s StudentAid.gov website). In each case, you must ensure that an individual with expertise in
the FSA programs is reasonably available shortly after the counseling to answer the grant recipient’s questions. (In the case of a grant recipient enrolled in a correspondence program or a
study-abroad program approved for credit at the home school, the grant recipient may be provided with written counseling materials within 30 days after he or she completes the program.)
It is the school’s responsibility to see that TEACH recipients receive TEACH Grant exit counseling before the student ceases attendance. If you require TEACH Grant recipients to complete exit
counseling on the Department’s StudentAid.gov website, you will receive reports from the Department on all students who have completed counseling. If a grant recipient doesn’t complete the exit
counseling session, you must ensure that exit counseling is provided either in person, through interactive electronic means, or by mailing written counseling materials (such as the PDF version of the
exit counseling program on the StudentAid.gov website) to the grant recipient’s last known address. In the case of unannounced withdrawals, you must provide this counseling within 30 days of learning
that a grant recipient has withdrawn from school (or from a TEACH Grant-eligible program).
The amount of a student’s TEACH Grant, in combination with any Pell Grant or other estimated financial assistance, may not exceed the student’s cost of attendance (COA). However, TEACH Grants may
replace the EFC for packaging purposes. See Chapter 7 of this volume for packaging rules.
Recalculating Teach Grants
Recalculating for changes in enrollment status
If a student’s enrollment status changes from one term to another within the same award year, you must recalculate the TEACH Grant award for the new payment period, taking into account any changes in
the COA.
If a student’s projected enrollment status changes during a payment period after the student has begun attendance in all of his or her classes for that payment period, you may (but are not required
to) establish a policy under which you recalculate such a student’s TEACH Grant award. Any such recalculations must take into account any changes in the COA. In the case of an undergraduate or
post-baccalaureate program of study, if such a policy is established, it must match your Pell Grant recalculation policy, and you must apply the policy to all students in the TEACH-eligible program.
If a student’s enrollment status changes during a payment period before the student begins attendance in all of his or her classes for that payment period, you must recalculate the student’s
enrollment status to reflect only those classes for which he or she actually began attendance.
TEACH Grant recalculations
Recalculating for changes in COA
If a student’s COA changes during the award year and his or her enrollment status remains the same, your school may, but is not required to, establish a policy under which you recalculate the
student’s TEACH Grant award. If you establish such a policy, you must apply it to all students in the program. | {"url":"https://fsapartners.ed.gov/knowledge-center/fsa-handbook/2020-2021/vol3/ch4-calculating-teach-grants","timestamp":"2024-11-10T16:19:05Z","content_type":"text/html","content_length":"99180","record_id":"<urn:uuid:c8b802c0-86b0-4049-8d31-a4f34bf8319c>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00353.warc.gz"} |
This is a text description of simpallc.gif. This figure illustrates the allocation of data values from a higher level in a dimension hierarchy to a lower level. In the figure are two sets of four
boxes. In each set one box is centered above three other boxes. The single box above the three lower boxes represents the higher, aggregate, level. In that box is the number 18. The three boxes below
each single box represent the lower, detail, level. From the bottom of each aggregate level box arrow point to its three detail level boxes. The set of boxes on the left represent an allocation that
uses the EVEN operator. The value 18 from the aggregate level is divided evenly between the three detail level boxes. In each detail level box is the number 6. The set of boxes on the right represent
an allocation that uses the PROPORTIONAL operator. The value 18 from the aggregate level is proportionately between the three detail level boxes. In the detail level box on the left is the number 2,
in the middle box is the number 6, and in the box on the right is the number 8. | {"url":"http://filibeto.org/sun/lib/nonsun/oracle/9.2.0.1.0/B10501_01/olap.920/a95298/img_text/simpallc.htm","timestamp":"2024-11-14T04:54:58Z","content_type":"text/html","content_length":"1548","record_id":"<urn:uuid:bea37839-c10c-489e-b4e8-6a915381a006>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00468.warc.gz"} |
6. Sīn, shīn, baṛī he, nūn, and nūn-e ġhunna
In Urdu, the sound s, or स, is represented by three different letters. By far the most common of these is sīn (the others will be introduced in Chapters 7 and 9). In its standard form, sīn has three
teeth, smaller than those of the be series and written close together. In the independent and final forms, these are attached to a bowl like that of lām:
Here are some words featuring sīn:
sabab ‘reason’
lěhsan ‘garlic’
sāl ‘year’
bas ‘enough’
sās ‘mother-in-law’
Sīn can be written in two ways. Sometimes, in place of the three teeth it may also be written with a long, gentle curve. This is especially common in handwriting. While this form is often used for
its aesthetic appeal, it also has a more practical function: to help distinguish two sīns when they appear side by side. Rather than write six small teeth in a row, you may optionally replace one of
the sets of teeth with a smooth line:
In the center of the image below, above the magazines, hangs an advertisement for the Express newspaper. The first sīn of Ekspres is written with teeth, but the second is not. Can you find another
sīn missing its teeth in the advertisement on the right-hand side? What word is written there?
In calligraphy, it is possible to stretch letters indefinitely in a technique called kashīda, as seen in bism, the first word below:
This long line can sometimes look like a a sīn without teeth, even when it isn’t one. In order to avoid confusion, calligraphers will sometimes write a small initial sīn above or below the actual sīn
. Can you spot this mark in the image below?
The letter shīn is used to write the sound sh or श (as well as retroflex ष, which is not distinguished in Urdu script). Once you’ve mastered the letter sīn, writing shīn is simple. Simply write a sīn
, and then add three dots above the main line. The dots should be arranged such that there are two at the bottom and the third nestled between them above:
ـش ـشـ شـ ش shīn
Here are some examples:
shādī ‘wedding’
tāsh ‘playing cards’
shukriya ‘thanks’
Like sīn, shin can be written in two ways: with teeth, and as an elongated curve. In either case, the dots should be centered above the letter.
The image below includes a long sīn and a toothed shīn, both marked with a small initial sīn / shīn below. Can you spot all of these on this bus’s destination board?
Baṛī he
In the previous chapter, you learned that there are two ways to write the sound h or ह. Usually, we use chhoṭī he, but sometimes we instead use its sibling, baṛī he. Baṛī he looks exactly like jīm,
but without any dot:
ـٮح ـٮحـ حـ ح baṛī he
बड़ी हे
Baṛī he only appears in words derived from Arabic. It is used to write a range of common words in Urdu, for instance:
hāl ‘condition’
muhabbat / mohabbat ‘love’
mahěl ‘palace’
rūh ‘soul’
masīh ‘Messiah’
What sweet treat does this package contain?
Halāl mārshmelo (halal marshmallows).
The letter nūn is used to represent the sound n or न. (It is also used for the retroflex ण, as well as the nasal sounds represented in Hindi script with the rarely used letters ङ and ञ.) In the
initial and medial forms, nūn looks like a be-series letter, with one dot above the tooth. In the final and independent forms, nūn takes on a bowl shape extending below the baseline, such that its
dot is located near the baseline itself:
Here are some words contaning nūn:
hinā ‘henna’
měhnat ‘labor’
nān ‘naan bread’
nānī ‘maternal grandmother’
gunāh ‘sin’
Hindi-Urdu verbs consist of a stem that is often followed by a suffix. For instance, the verb bannā ‘to become’ contains the stem ban and the suffix -nā, which indicates the infinitive form. With the
same stem, we can make other forms of the verb, like bantā ‘becomes,’ banegā ‘will become,’ banā ‘became,’ and so forth.
In Chapter 3, we introduced the tashdīd and said that consonants are not written twice except when there is a vowel in between them. However, in both Hindi and Urdu scripts, verb suffixes are an
exception to this rule. Because a suffix is a separate unit of meaning that is added on to the stem, when it begins with the last letter of the stem, they are both written, even though there is no
vowel between them. Thus we have:
bannā बनना ‘to become’
sunnā सुनना ‘to hear’
This rule is not exclusive to nūn:
وہ ہمیشہ جیتتا ہے۔
वह हमेशा जीतता है।
Wo hamesha jīttā hai.
He always wins.
Nasalization with nūn-e ġhunna
Nasalization refers to that quality of pronouncing a vowel sound through your nose, marked in Hindi script by the chandrabindu, as in the word hāñ हाँ ‘yes.’ In Urdu script, nasalization is represented
by a derivative form of the letter nūn called nūn-e ġhunna (or nūn ġhunna), ‘nasal nūn’:
ـں ـنـ نـ ں nūn-e ġhunna (nūn-ġhunna)
नून-ए-ग़ुन्ना (नून-ग़ुन्ना)
In the final and independent forms, nūn-e ġhunna looks like nūn, but without any dot:
kareñ ‘would do’
yahañ ‘here’
maiñ ‘I’ / meñ ‘in’
nahīñ ‘no’
haiñ ‘are’
běhneñ ‘sisters’
In the initial and medial forms, nūn-e ġhunna looks identical to the regular nūn (though it is occasionally marked with a sukūn or a small semicircular diacritic):
sāñs ‘breath’
bāñsurī ‘flute’
muñh ‘mouth’
As we mentioned in Chapter 4, chhoṭī he and do-chashmī he can be used interchangeably. A do-chashmī he often takes the place of a chhoṭī he when it follows either a nūn-e ġhunna or a nasal consonant
(n or m) with a sukūn—in terms of Hindi script, a chandrabindu or a nasal half-letter. In other words, it is as if we are writing an aspirated mh or nh.
These are some words that you might see spelled in this alternate way:
muñh मुँह ‘mouth’
tumhārā तुम्हारा ‘your’
unheñ उन्हें ‘to them’
انھوں نے
unhoñ ne उन्होंने / उन्हों ने ‘they’ (with a perfect transitive verb)
The use of nūn-e ġhunna can sometimes be a bit counterintuitive, especially when a nasal vowel comes before a sound like p or b and thus you might expect to use a mīm:
sañp ‘snake’
kāñpnā, ‘to tremble’
One place where a final nūn-e ġhunna is used in the middle of a word is in a future-tense verb. This is because the gā / ge / gī suffix that marks these verbs is usually written as a separate word:
کتا چلے گا
kuttā chalegā ‘the dog will go’
بلی چلے گی
billī chalegī ‘the cat will go’
کتے چلیں گے
kutte chaleñge ‘the dogs will go’
بلیاں چلیں گی
billiyāñ chaleñgī ‘the cats will go’
Some sounds can be represented by multiple Urdu letters.
Nasal vowels are written with nūn-e ġhunna, which looks like an ordinary nūn in the initial and medial positions.
In this chapter, we introduced these letters:
Letter Name Sound
س sīn सीन s स
ش shīn शीन sh श
ح baṛī he बड़ी हे h ह
ن nūn नून n न
ں nūn-e ġhunna (nūn-ġhunna) नून-ए-ग़ुन्ना (नून-ग़ुन्ना) ñ ँ
The rounded bottom part of a letter, extending below the baseline. Bowls appear in the independent and final forms of letters including lām, nūn, and sīn.
A way of writing elongated letters, traditionally used for justifying lines of text (rather than by adjusting white space, as is typical in English) and for ornamental purposes. Literally “pulled.”
The portion of a verb that carries its basic meaning, like dekh ‘see,’ khā ‘eat,’ etc.
An infinitive is a verb in its basic form, composed of a stem with the suffix -nā, for example karnā ‘to do,’ honā ‘to be,’ dekhnā ‘to see,’ etc. In Hindi-Urdu, the infinitive can also act as a noun,
e.g. karnā ‘doing,’ honā ‘being,’ dekhnā ‘seeing,’ and so forth.
The perfect tense is used for actions that are completed, e.g. khāyā, ga’ī, hu’e, etc. The perfect tense takes a special form when it occurs with transitive verbs.
Transitive verbs are those with direct objects. In the perfect tense, Hindi-Urdu transitive verbs (apart from a small number of exceptions, like lānā) match the number and gender of the grammatical
object rather than the subject, and the subject is followed by the postposition ne. | {"url":"https://openbooks.library.northwestern.edu/zerozabar/chapter/sin-shin-bari-he-nun-nun-ghunna/","timestamp":"2024-11-04T14:29:44Z","content_type":"text/html","content_length":"141910","record_id":"<urn:uuid:4302c54b-8412-465b-816a-fe609080fa2e>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00621.warc.gz"} |
6th grade math teachers worksheets on percents
Algebra Tutorials!
6th grade math teachers worksheets on percents
Related topics:
Home application of linear algebra in everyday life | online calculator for solving quadratics | grade nine slope lesson | mathcad inequality constraint system of nonlinear
Solving Quadratic equations | computer lessons with completing the square | sciencetific symbols | convert int biginteger java
Equations by Completing
the Square
Graphing Logarithmic Author Message
Division Property of Jan Leed Posted: Wednesday 27th of Dec 08:59
Exponents I'm having immense difficulty finding the logic behind the problem regarding 6th grade math teachers worksheets on percents. Can somebody please
Adding and Subtracting assist me to know how to come up with a detailed answer and explanation about 6th grade math teachers worksheets on percents especially in topic of
Rational Expressions multiplying matrices? I was taught how to do this before but now I forgot and confused how to solve it. I find it complicated to understand it on my
With Like Denominators own so I believe I need assistance since I think I can’t do this on my own . If someone knows about 6th grade math teachers worksheets on percents
Rationalizing the Registered: can you please help me? Thanks!
Denominator 10.02.2003
Multiplying Special From: Manchester,
Polynomials United Kingdom
Solving Linear Systems
of Equations by
Elimination Jahm Xjardx Posted: Thursday 28th of Dec 13:35
Solving Systems of Due to health reasons you may have missed a few lectures at school, but what if I can you can simulate your classroom, in the place where you live?
Equation by Substitution In fact, right on the computer that you are working on? Each one of us has missed some lectures at some point or the other during our life, but
and Elimination thanks to Algebrator I’ve never been left behind . Just like a teacher would explain in the class, Algebrator solves our queries and gives us a
Polynomial Equations detailed description of how it was solved . I used it basically to get some help on 6th grade math teachers worksheets on percents and quadratic
Solving Linear Systems Registered: inequalities. But it works well for just about everything you can think of .
of Equations by Graphing 07.08.2005
Quadratic Functions From: Odense,
Solving Proportions Denmark, EU
Parallel and
Perpendicular Lines
Simplifying Square Roots
Simplifying Fractions Admilal`Leker Posted: Thursday 28th of Dec 17:22
Adding and Subtracting I always use Algebrator to help me with my math homework . I have tried several other software programs but so far this is the best I have seen. I
Fractions guess it is the detailed way of explaining the solution to problems that makes the whole process appear so easy. It is indeed a very good piece of
Adding and Subtracting software and I can vouch for it.
Solving Linear Equations Registered:
Inequalities in one 10.07.2002
Variable From: NW AR, USA
Recognizing Polynomial
Equations from their
Scientific Notation dn6tish Posted: Thursday 28th of Dec 21:52
Factoring a Sum or I am not trying to run away from my problems . I do admit that sleeping off is not a solution to it either. Please let me know where I can find this
Difference of Two Cubes piece of software.
Solving Nonlinear
Equations by
Substitution Registered:
Solving Systems of 14.04.2003
Linear Inequalities From:
Arithmetics with
Finding the Equation of
an Inverse Function TC Posted: Saturday 30th of Dec 17:48
Plotting Points in the Well, you don’t have to wait any longer. Go to https://algebra-calculator.com/simplifying-fractions.html and get yourself a copy for a very nominal
Coordinate Plane price. Good luck and happy learning!
The Product of the Roots
of a Quadratic
Powers Registered:
Solving Quadratic 25.09.2001
Equations by Completing From: Kµlt °ƒ Ø,
the Square working on my time
sxAoc Posted: Saturday 30th of Dec 19:47
I recommend using Algebrator. It not only assists you with your math problems, but also displays all the necessary steps in detail so that you can
enhance the understanding of the subject.
From: Australia
6th grade math teachers worksheets on percents
Related topics:
Home application of linear algebra in everyday life | online calculator for solving quadratics | grade nine slope lesson | mathcad inequality constraint system of nonlinear
Solving Quadratic equations | computer lessons with completing the square | sciencetific symbols | convert int biginteger java
Equations by Completing
the Square
Graphing Logarithmic Author Message
Division Property of Jan Leed Posted: Wednesday 27th of Dec 08:59
Exponents I'm having immense difficulty finding the logic behind the problem regarding 6th grade math teachers worksheets on percents. Can somebody please
Adding and Subtracting assist me to know how to come up with a detailed answer and explanation about 6th grade math teachers worksheets on percents especially in topic of
Rational Expressions multiplying matrices? I was taught how to do this before but now I forgot and confused how to solve it. I find it complicated to understand it on my
With Like Denominators own so I believe I need assistance since I think I can’t do this on my own . If someone knows about 6th grade math teachers worksheets on percents
Rationalizing the Registered: can you please help me? Thanks!
Denominator 10.02.2003
Multiplying Special From: Manchester,
Polynomials United Kingdom
Solving Linear Systems
of Equations by
Elimination Jahm Xjardx Posted: Thursday 28th of Dec 13:35
Solving Systems of Due to health reasons you may have missed a few lectures at school, but what if I can you can simulate your classroom, in the place where you live?
Equation by Substitution In fact, right on the computer that you are working on? Each one of us has missed some lectures at some point or the other during our life, but
and Elimination thanks to Algebrator I’ve never been left behind . Just like a teacher would explain in the class, Algebrator solves our queries and gives us a
Polynomial Equations detailed description of how it was solved . I used it basically to get some help on 6th grade math teachers worksheets on percents and quadratic
Solving Linear Systems Registered: inequalities. But it works well for just about everything you can think of .
of Equations by Graphing 07.08.2005
Quadratic Functions From: Odense,
Solving Proportions Denmark, EU
Parallel and
Perpendicular Lines
Simplifying Square Roots
Simplifying Fractions Admilal`Leker Posted: Thursday 28th of Dec 17:22
Adding and Subtracting I always use Algebrator to help me with my math homework . I have tried several other software programs but so far this is the best I have seen. I
Fractions guess it is the detailed way of explaining the solution to problems that makes the whole process appear so easy. It is indeed a very good piece of
Adding and Subtracting software and I can vouch for it.
Solving Linear Equations Registered:
Inequalities in one 10.07.2002
Variable From: NW AR, USA
Recognizing Polynomial
Equations from their
Scientific Notation dn6tish Posted: Thursday 28th of Dec 21:52
Factoring a Sum or I am not trying to run away from my problems . I do admit that sleeping off is not a solution to it either. Please let me know where I can find this
Difference of Two Cubes piece of software.
Solving Nonlinear
Equations by
Substitution Registered:
Solving Systems of 14.04.2003
Linear Inequalities From:
Arithmetics with
Finding the Equation of
an Inverse Function TC Posted: Saturday 30th of Dec 17:48
Plotting Points in the Well, you don’t have to wait any longer. Go to https://algebra-calculator.com/simplifying-fractions.html and get yourself a copy for a very nominal
Coordinate Plane price. Good luck and happy learning!
The Product of the Roots
of a Quadratic
Powers Registered:
Solving Quadratic 25.09.2001
Equations by Completing From: Kµlt °ƒ Ø,
the Square working on my time
sxAoc Posted: Saturday 30th of Dec 19:47
I recommend using Algebrator. It not only assists you with your math problems, but also displays all the necessary steps in detail so that you can
enhance the understanding of the subject.
From: Australia
Solving Quadratic
Equations by Completing
the Square
Graphing Logarithmic
Division Property of
Adding and Subtracting
Rational Expressions
With Like Denominators
Rationalizing the
Multiplying Special
Solving Linear Systems
of Equations by
Solving Systems of
Equation by Substitution
and Elimination
Polynomial Equations
Solving Linear Systems
of Equations by Graphing
Quadratic Functions
Solving Proportions
Parallel and
Perpendicular Lines
Simplifying Square Roots
Simplifying Fractions
Adding and Subtracting
Adding and Subtracting
Solving Linear Equations
Inequalities in one
Recognizing Polynomial
Equations from their
Scientific Notation
Factoring a Sum or
Difference of Two Cubes
Solving Nonlinear
Equations by
Solving Systems of
Linear Inequalities
Arithmetics with
Finding the Equation of
an Inverse Function
Plotting Points in the
Coordinate Plane
The Product of the Roots
of a Quadratic
Solving Quadratic
Equations by Completing
the Square
Author Message
Jan Leed Posted: Wednesday 27th of Dec 08:59
I'm having immense difficulty finding the logic behind the problem regarding 6th grade math teachers worksheets on percents. Can somebody please assist me to know how to come up
with a detailed answer and explanation about 6th grade math teachers worksheets on percents especially in topic of multiplying matrices? I was taught how to do this before but
now I forgot and confused how to solve it. I find it complicated to understand it on my own so I believe I need assistance since I think I can’t do this on my own . If someone
knows about 6th grade math teachers worksheets on percents can you please help me? Thanks!
From: Manchester,
United Kingdom
Jahm Xjardx Posted: Thursday 28th of Dec 13:35
Due to health reasons you may have missed a few lectures at school, but what if I can you can simulate your classroom, in the place where you live? In fact, right on the
computer that you are working on? Each one of us has missed some lectures at some point or the other during our life, but thanks to Algebrator I’ve never been left behind . Just
like a teacher would explain in the class, Algebrator solves our queries and gives us a detailed description of how it was solved . I used it basically to get some help on 6th
grade math teachers worksheets on percents and quadratic inequalities. But it works well for just about everything you can think of .
From: Odense,
Denmark, EU
Admilal`Leker Posted: Thursday 28th of Dec 17:22
I always use Algebrator to help me with my math homework . I have tried several other software programs but so far this is the best I have seen. I guess it is the detailed way
of explaining the solution to problems that makes the whole process appear so easy. It is indeed a very good piece of software and I can vouch for it.
From: NW AR, USA
dn6tish Posted: Thursday 28th of Dec 21:52
I am not trying to run away from my problems . I do admit that sleeping off is not a solution to it either. Please let me know where I can find this piece of software.
TC Posted: Saturday 30th of Dec 17:48
Well, you don’t have to wait any longer. Go to https://algebra-calculator.com/simplifying-fractions.html and get yourself a copy for a very nominal price. Good luck and happy
From: Kµlt °ƒ Ø,
working on my time
sxAoc Posted: Saturday 30th of Dec 19:47
I recommend using Algebrator. It not only assists you with your math problems, but also displays all the necessary steps in detail so that you can enhance the understanding of
the subject.
From: Australia
Posted: Wednesday 27th of Dec 08:59
I'm having immense difficulty finding the logic behind the problem regarding 6th grade math teachers worksheets on percents. Can somebody please assist me to know how to come up with a detailed
answer and explanation about 6th grade math teachers worksheets on percents especially in topic of multiplying matrices? I was taught how to do this before but now I forgot and confused how to solve
it. I find it complicated to understand it on my own so I believe I need assistance since I think I can’t do this on my own . If someone knows about 6th grade math teachers worksheets on percents can
you please help me? Thanks!
Posted: Thursday 28th of Dec 13:35
Due to health reasons you may have missed a few lectures at school, but what if I can you can simulate your classroom, in the place where you live? In fact, right on the computer that you are working
on? Each one of us has missed some lectures at some point or the other during our life, but thanks to Algebrator I’ve never been left behind . Just like a teacher would explain in the class,
Algebrator solves our queries and gives us a detailed description of how it was solved . I used it basically to get some help on 6th grade math teachers worksheets on percents and quadratic
inequalities. But it works well for just about everything you can think of .
Posted: Thursday 28th of Dec 17:22
I always use Algebrator to help me with my math homework . I have tried several other software programs but so far this is the best I have seen. I guess it is the detailed way of explaining the
solution to problems that makes the whole process appear so easy. It is indeed a very good piece of software and I can vouch for it.
Posted: Thursday 28th of Dec 21:52
I am not trying to run away from my problems . I do admit that sleeping off is not a solution to it either. Please let me know where I can find this piece of software.
Posted: Saturday 30th of Dec 17:48
Well, you don’t have to wait any longer. Go to https://algebra-calculator.com/simplifying-fractions.html and get yourself a copy for a very nominal price. Good luck and happy learning!
Posted: Saturday 30th of Dec 19:47
I recommend using Algebrator. It not only assists you with your math problems, but also displays all the necessary steps in detail so that you can enhance the understanding of the subject. | {"url":"https://algebra-calculator.com/algebra-calculator-program/ratios/6th-grade-math-teachers.html","timestamp":"2024-11-14T17:15:48Z","content_type":"text/html","content_length":"96972","record_id":"<urn:uuid:5d89c775-587b-4871-9ba9-e39ed3569d2a>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00631.warc.gz"} |
7.3 Design of slopes
7.3.1 Methods of design of soil slopes
The designer may use some or all of the following design methods:
• a) limit-equilibrium methods (see BS EN 1997-1:2004, 11.5.1, which requires horizontal interslice forces to be assumed unless horizontal equilibrium is checked; this excludes Janbu's original
method and the Swedish circle [aka Fellenius] method, but allows, for example, Bishop's, Janbu's simplified and modified and Sarma's methods);
• b) numerical methods (see BS EN 1997-1:2004, 2.4.1(12));
• c) physical modelling (see BS EN 1997-1:2004, 2.6);
• d) prescriptive measures (see BS EN 1997-1:2004, 2.5);
• e) observational method (see BS EN 1997-1:2004, 2.7 and CIRIA R185 [16]);
• f) stability charts;
• g) infinite slope method.
7.3.2 Methods of design of rock slopes
Unlike soil slopes, the design of rock slopes is dominated by discontinuities, and recognized references such as Hoek and Bray (in Wyllie and Mah [26]) and TRL [27] should be consulted.
The design should consider:
• a) the stability of the rock mass, which in most cases is governed by conditions in the joint system of the mass rather than by the strength of the intact rock; an assessment is required of the
discontinuities within the rock mass, including any infilling;
• b) drainage requirements to manage groundwater, particularly where preferential groundwater flow is most likely (e.g. along soil/rock interface, discontinuities, permeable zones);
• c) local experience or exposures in similar strata;
• d) standard details required to deal with all the adverse conditions that can be reasonably anticipated (e.g. rock bolting, dentition work, drainage); and
• e) potential deterioration of the rock mass or discontinuities due to weathering effects during the design life of the excavated face.
In rock slope design it is particularly important that the designer should assess the ground conditions anticipated within an excavation (including potential unfavourable conditions), the proposed
works best suited to deal with those conditions, and the form of inspection and design check as part of the works. For new rock cuttings, a trial excavation should usually be made to enable a check
to be made of the design assumptions prior to cutting the face to the required finished position.
Weak, heavily weathered rocks can exhibit engineering characteristics intermediate between those of a soil and those of a rock; in cases of doubt, separate analyses of slope stability should be made
assuming that the material behaves either as a soil or as a rock.
7.3.3 Factors of safety and partial factors
The verification of the overall stability of slopes should be carried out in accordance with BS EN 1997-1:2004.
The overall stability of slopes should be checked against DA1 Combination 2. For completeness, DA1 Combination 1 should also be checked if the designer considers that the loading applied to the slope
(other than the mass of the ground in the slope) might control the failure mechanism rather than the ground strength parameters [see BS EN 1997-1:2004, 2.4.7.3.4.2(3)].
COMMENTARY ON 7.3.3
BS EN 1997-1:2004, 11.5.1 details the requirements for determining the overall stability of slopes and BS EN 1997-1:2004, 2.4.7.3.4 sets out the Design Approaches which are to be applied. The
National Annex adopts Design Approach 1 (DA1), which requires verification that the limit state of rupture or excessive deformation will not occur with either of the following combinations of sets of
partial factors.
Combination 1: A1 "+" M1 "+" R1 Combination 2: A2 "+" M2 "+" R1 Where "+" implies "to be combined with".
In Combination 1, partial factors in excess of unity are applied to unfavourable actions or the effects of actions whereas in Combination 2, the inverse of partial factors exceeding unity are applied
to the soil parameters. This has the effect of increasing the effect of actions in Combination 1 and reducing the ground strength in Combination 2. The basic equations that govern are:
• F[d] is the design value of an action;
• γ[F] is the partial factor for that action;
• F[rep] is the representative value for that action;
• X[dis] the design value for a material property;
• X[k] is the characteristic value for that material property; and
• γ[M] is the partial factor for that material property.
The partial factors that should be applied to actions and to ground strength parameters are set by NA to BS EN 1997-1:2004, which are given in Table 4, Table 5 and Table 6. However,
NA to BS EN 1997-1:2004 does not provide partial factors for actions for the specific situation of earthworks. In the absence of these, the values in Table 4 are recommended [based on the values for
buildings given in NA to BS EN 1990:2002+A1, Table NA.A1.2 (A)]. Reference should be made to the current version of NA to BS EN 1997-1 to ensure the correct partial factors are used for design.
Table 4 Partial factors on actions or the effects of actions
Action Symbol Set
A1 A2
Permanent Unfavourable γ[G] 1,35 1,0
Favourable 1,0 1,0
Variable Unfavourable γ[Q] 1,5 1,3
Favourable 0 0
Table 5 Partial factors for soil parameters
Soil parameter Symbol Set
M1 M2
Angle of shearing resistance ^A) γ[ϕ'] 1,0 1,25
Effective cohesion γ[c'] 1,0 1,25
Undrained shear strength γ[cu] 1,0 1,4
Unconfined strength γ[qu] 1,0 1,4
Weight density γ[g] 1,0 1,0
^A) Factor applied to tan ϕ' (see text of this clause for partial factor applied to residual angle of shearing resistance).
Table 6 Partial resistance factors for slopes and overall stability
Resistance Symbol Set
Earth resistance γ[R;e] 1,0
COMMENTARY ON 7.3.3 (continued)
Combination 1 involves applying partial factors to actions or the effects of actions whilst using unfactored values for the soil parameters and earth resistance. This approach is not usually relevant
for checking the overall stability of a slope where earth is the main element providing resistance, since structural strengths do not provide resistance against overall stability failure and failure
is controlled by uncertainty in the ground strength rather than uncertainty in the actions.
In addition the treatment of actions due to gravity, loads and water is difficult since these loads might be unfavourable in part of the sliding mass but favourable in another part. In a traditional
analysis of a circular failure surface, part of the slope mass is producing a positive driving moment (i.e. it is unfavourable) and part of the slope mass is producing a negative driving moment (i.e.
it is favourable) and the moments produced by the two parts depend on the position of the point about which moment equilibrium is checked. The application of different partial factors to each part of
the slope introduces scope for confusion and requires a degree of complexity of analysis that is not readily available and not justified given the nature of the problem.
For this reason, a note to 2.4.2 of BS EN 1997-1:2004 states "Unfavourable (or destabilizing) and favourable (or stabilizing) permanent actions may in some situations be considered as coming from a
single source. If they are considered so, a single partial factor may be applied to the sum of these actions or to the sum of their effects." This note, commonly referred to as the "single-source
principle", allows the same partial factor to be applied to stabilizing and destabilizing actions. When using Combination 1, it is recommended that the partial factor for the unfavourable action of
the soil is applied to the weight density of the soil and the effect of this application can be summarized as follows.
• • In an effective stress analysis, the effect of the partial factor is to increase the destabilizing action and to increase simultaneously the shearing resistance of the soil, which cancels the
effect of the partial factor.
• • In a total stress analysis, the increase in weight density increases the destabilizing action without increasing the shearing resistance of the soil. However, a higher partial factor is applied
to the undrained strength in Combination 2 than to the permanent destabilizing action in Combination 1.
In both cases, Combination 1tends to be less critical than Combination 2 in almost all design situations. (Exceptions might occur when extremely large variable actions apply or the soil strength is
extremely low.) Bond and Harris [28] discuss the way in which the single-source principle should be applied to slopes and embankments and show that Combination 2 results in an equivalent global
factor of safety of about 1.25 for typical situations where an effective stress analysis is used.
If the single-source principle is not applied, then a special procedure has to be followed, if using commercially available software, in order to apply different factors to stabilizing and
destabilizing actions. Frank et al [5] describe one such procedure, but by ignoring the single-source principle, Combination 1 becomes more critical than Combination 2 in most design situations using
an effective stress analysis and results in an equivalent global factor of safety of about 1.35. However, Frank et al [5] recommend that Combination 2 normally be used for checking the overall
stability of earthworks since the stability is governed by the shear strength of the soil rather than the application of the load of the earthworks.
Subclause 2.4.7.3.4.2(3) of BS EN 1997-1:2004 states that, in circumstances where it is obvious that one of the two combinations governs the design, calculations for the other combination need not be
carried out, but the designer needs to be sure that this is the case (e.g. based on past experience of similar designs). Therefore it is acceptable to base designs on Combination 2 alone (invoking
the single-source principle) for many typical situations.
Where there is significant uncertainty about the density of the ground a sensitivity analysis should be undertaken [see BS EN 1997-1:2004, 11.5.1(12)].
Guidance on the use of advanced numerical methods in conjunction with the partial factors given in BS EN 1997-1:2004 is provided by Frank et al [5]; however, the designer should consider the
relevance of such methods to the problem under consideration before embarking on advanced design since the overall stability of most routine slopes can be verified using limit-equilibrium methods.
The partial factors normally used for overall stability analyses may not be appropriate for slopes with pre-existing failure surfaces [BS EN 1997-1:2004, 11.5.1(8)], in which case the following
approaches are relevant.
• Where the soil parameters for pre-existing failure surfaces are determined by back analysis partial factors of unity should be used for actions and the effects of actions, soil parameters and
earth resistance since the objective in this case is to determine the value of the mobilized shear strength along the pre-existing failure surface.
• In the case where the residual strength of the soil is used for design purposes (whether determined from back analysis, laboratory or in-situ testing or from published data) Design Approach 1,
Combination 2 is likely to govern the overall stability of the slope. BS EN 1997-1:2004, 11.5.1(8) states that partial factors normally used for overall stability need not be appropriate for the
analysis of existing failed slopes therefore lower values of the partial factors for ground strength parameters than those given in NA to BS EN 1997-1:2004 for Set M2 (i.e. the factors used in
Design Approach 1 Combination 2) may be applied to residual strength. The partial factor used with the residual angle of shearing resistance should be chosen with due consideration to the
confidence level of the data and the consequences of subsequent failure of the slope. Usually it should not be necessary for the partial factor applied to the residual angle of shearing
resistance to exceed 1.1 provided the effective cohesion used in conjunction with that angle is set to zero.
For any slope where the consequences of slope failure are abnormally high the selection of characteristic values for the soil parameters should reflect the increased risk (see 7.4) in addition to
other considerations listed in BS EN 1997-1:2004, 2.4.5.2(4) and a very cautious value might have to be chosen for the characteristic value. Alternatively, consideration should be given to increasing
the partial factors on actions or the effects of actions and/or those for soil parameters.
NOTE The designer is referred to Frank et al [5], Bond and Harris [28] and CIRIA C641 [7] for examples of calculations and further guidance on design to EC7 principles. These references give worked
examples of analysis by rotational, wedge and infinite slope methods, consider analysis by computer software or stability charts, and also identify some areas where differences can be expected
relative to conventional global factor of safety methods of analysis.
7.3.4 Seismic effects
The designer should assess the potential seismicity of the region and, where appropriate, the requirements of BS EN 1998-5.
NOTE It is not normal to consider seismic effects for Category 1 and Category 2 structures in the UK. | {"url":"https://geotechnicaldesign.info/bs6031-2009/g7-3.html","timestamp":"2024-11-09T08:10:10Z","content_type":"text/html","content_length":"55777","record_id":"<urn:uuid:822ffdd0-3690-492d-a654-98e0325e07ed>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00530.warc.gz"} |
How to Calculate the Cost of Capital for Your BusinessHow to Calculate the Cost of Capital for Your Business | Cleverism
How to Calculate the Cost of Capital for Your Business
by Martin Luenendonk
• How to Calculate the Cost of Capital for Your Business
Companies and investment funds are currently sitting on a lot of money. But before they start putting this capital into new use, it is important to understand more about the cost of financing
different investments offer to their business. In order to do so, businesses must calculate the cost of capital.
But what is the cost of capital and how can companies calculate it? This guide will answer these important questions and help you understand why cost of capital is among the most important business
formulas you’ll need to learn about. You’ll also be able to understand the common pitfalls and limitations of calculating this important figure for your business.
WHAT DOES ‘COST OF CAPITAL’ MEAN?
The definition of cost of capital simply means the cost of funds the company uses to fund and finance its operations. The cost of capital is often divided into two separate modes of financing: debt
and equity.
Cost of capital tells the company its hurdle rate. The hurdle rate refers to the minimum rate of return the company must achieve to be profitable or to generate value.
Each company has its own cost of capital. Different factors influence the cost of capital and these include things such as the operating history of the business, its profitability and credit
The figure is one of the most essential parts of a business’ financing strategy, as it can help the company to make better funding and investment decisions and thus boost its overall financial
In case the company is solely financed through equity, the cost of capital would refer to the cost of equity. On the other hand, companies funded by debt alone have cost of capital refer to the cost
of debt.
As most companies rely on a combination of debt and equity, their overall cost of capital is derived from a weighted average of all capital sources. This refers to the average cost of capital (WACC).
The difference between cost of equity and cost of debt
If the company’s only source has been equity put in by the company’s owners or shareholders, then you can simply calculate the cost of capital by analyzing the cost of equity. The cost of equity then
represents the compensation the market demands in exchange for the company’s assets.
On the other hand, the cost of debt refers to situations where the company has funded itself through debt alone. This would mean the company has financed all of its operations simply by lending from
creditors. By calculating the cost of debt, you’ll receive the cost of capital.
The cost of debt reveals the effective rate the company should pay its current debt. Since interest is also added into the calculation, the cost of debt can either be measured before-tax or
The reason companies are aiming for a balanced mixture of debt and equity financing is to descrease the overall cost of capital in both cases and what it means for the business’ finances. For
example, while debt financing is more tax-efficient to equity financing, high levels of debt can result in higher leverage, which means higher interest rates due to increased risk. Therefore, a
mixture of both financing sources often provides the lowest cost of capital.
The definition of weighted average cost of capital (WACC)
It gives a proportional weight to the different costs of capital, such as equity and debt, to derive a weighted average cost. Each capital component will be multiplied by its proportional weight and
the sums will be added together.
When companies refer to the cost capital, they often would have calculated it based of the WACC method. The following sections will look at the calculations methods in more detail, but here’s a quick
example of what WACC means.
Consider that a business has a lender, which requires a 10% return on its money. Furthermore, the shareholders of the business require a further minimum of a 20% on their investments. On average
then, the company’s capital must have a return of 15% to satisfy both the debt and equity holders, meaning the WACC or cost of capital is 15%.
This means the company would need to invest in projects that would provide an annual return of 15% in order to continue paying back to both their shareholders and creditors.
Before we look at the formulas to calculate the cost of capital in more detail, it is important to understand why it is essential to do the maths. As mentioned briefly above, the cost of capital can
be an essential part of a business’ financial decision-making.
Since cost of capital provides the business with the minimum rate of return it needs on its investments, it is an essential part of budgeting decisions. By knowing the cost of capital, the business
can make better decisions on its future investments and other such financing options.
For example, it can help the business to find projects that will generate appropriate gains for the business. On the other hand, it can prevent the business from making an investment, which wouldn’t
provide quick enough returns for the company.
Therefore, a cost of capital reveals the business plenty about the type and value of its past and future investments. If a business doesn’t know the rate of return or the cost of financing its
operations, it can’t expect much business success.
In addition, it’ll help better attract new investors for the business, as they are able to understand the kind of rate of return they will receive. It also ensures the business doesn’t go after
creditors or investors it cannot repay at the current time.
Overall, understanding the cost of capital will boost the business’ financial decision-making. Because the cost of capital is used to design the market fluctuations, it can help build better
financial structures.
In some instances, businesses even use it to better understand financial performance and to evaluate whether the management is performing well enough.
Now that you understand the definition of cost of capital and the importance of calculating it, it’s time to look at the calculating methods.
First, we’ll go through the formulas for calculating both the cost of equity and debt, as they’ll be used in the final calculations of WACC. Naturally, if the business only uses either debt or equity
alone, you can also use the formulas as the basis for calculating the cost of capital.
Calculating the cost of debt
First, lets look at how you can calculate the cost of debt. Debt in this formula includes all forms of debt the company uses in order to finance its operations. These could be various bonds, loans
and other such forms of debt.
As mentioned earlier, there are two formulas for calculating the cost of debt. This is because it deals with interest, which can be deducted from tax payments. Thus, the alternatives are to calculate
the cost of debt either before- or after-tax. Generally, the after-tax cost is more widely used.
The before-tax rate can be calculated by two different methods. First, you can calculate it by multiplying the interest rate of the company’s debt by the principal. For instance, a $100,000 debt bond
with 5% pre-tax interest rate, the calculation would be: $100,000 x 0.05 = $5,000.
The second method uses the after-tax adjusted interest rate and the company’s tax rate.
Even if you use the after-tax rate, you’ll still need the above before-tax rate. The formula for calculating the after-rate tax is:
Cost of debt (after-tax rate) = before-tax rate * (1 – marginal tax rate)
Keep in mind the before-tax rate is also often referred to as the yield-to-maturity on long-term debt.
Calculating the cost of equity
There are also two ways of calculating the cost of equity: the more traditional dividend capitalization model and the more modern capital asset pricing model (CAPM).
The dividend capitalization model uses the following formula:
Cost of equity = (dividends per share [for next year] / current market value of stock) + growth rate of dividends
More recently, many companies have started to the use the CAPM method. Under this method, the idea is that investors need a minimum rate of return, which is equal to return from a risk-free
investment, as well as a return for bearing extra risk.
The formula is as follows:
Cost of equity = risk free rate + beta [i.e. risk measure] * (expected market return – risk free rate)
Calculating WACC
If the company has used different methods of financing, then the cost of capital is calculated by the weighted average cost of capital. The above formulas are also needed in this method.
The method for calculating WACC is often expressed in the following formula:
WACC = percentage of financing that is equity * cost of equity + percentage of financing that is debt * cost of debt * (1 – corporate tax rate)
In order to calculate the percentage of financing that is equity, you need the following formula:
Percentage of financing that is equity = market value of the firm’s equity / total market value of the firm’s financing (equity and debt)
To calculate the percentage of financing that is debt, you can use the following formula:
Percentage of financing that is debt= market value of the firm’s debt / total market value of the firm’s financing (equity and debt)
The WACC will increase if the beta (risk measure) and the rate of return on equity increase. This is because a growing WACC denotes a drop in valuation and a growth in risk.
You can also find out the above information from this informative YouTube video:
To make the above formulas a bit less daunting, here’s an example calculation of WACC. The below calculation is a rather simplified version of the different factors that might influence the rates
used in the calculation. To ensure you come up with the most accurate figure for the cost of capital, you also need to check out the common problems in calculating it in the following section.
In our example, the crucial figures in WACC are as follows:
The company’s total equity = $10,000
The company’s total debt = $3,000
The cost of equity = 12.5%
The cost of debt = 6%
The tax rate = 28%
Therefore, the WACC will be calculated by solving the formula:
10,000/13,000 * 12.5% + 3,000/13,000 * 6%*(1-28%) = 10.84%
Therefore, the cost of capital for the business is 10.84%.
In reality, calculating the different aspects isn’t quite as quick and straightforward. Therefore, most companies use different online and offline tools as a helpful guide for calculating the
For example, you can find Excel-files, which allow you to simply add the different figures into the file and receive the final rate in an instant.
While it is essential to calculate the cost of capital for your business, you need to be aware of some of the pitfalls as well as limitations behind this method. A survey by the Association for
Financial Professionals recently found that many companies don’t use universal methods for calculating the cost of capital and the assumptions many make can lead to distorted estimations of the real
cost of capital. Naturally, this can have devastating consequences, as it might mean the company makes investment decisions based on incorrect information.
In order to avoid these issues with your calculation, here are some of the most common problems you should try to avoid.
Using the wrong investment time horizon
The first issue often comes when companies select their forecast periods for variables such as cash flow. The survey, mentioned above, found that companies’ estimates could range from five years to
15-year horizon!
Naturally, different companies can expect investments will live a different time span. But the crucial thing to remember is how the chosen time horizon should reflect the kind of project in question,
instead of simply being a standard time period.
If you are calculating the cost of capital for a specific investment project, remember to keep this in mind. Evaluating the nature of the project is a crucial part of success.
Trouble selecting the right risk-free rate
As you remember, the cost of equity formula dealt with risk-free rates. The differences in calculations come from the fact that there aren’t any universal risk-free rates available.
In the US, many use the US Treasury’s rates as the benchmark, but since these also come in different time horizons, the final calculations can change a lot depending on which time ratio you choose to
use. For example, the 90-day Treasury note could yield 0.05%, with the 10-year note yielding 2.25%.
This could mean two similar types of businesses have very different cost of equity, solely because they used a different risk-free rate.
While it isn’t necessarily easy to overcome this issue, it is good to keep in mind, especially if you are an investor. Furthermore, you should consider mentioning the risk-free rate in the footnotes
to ensure you always know what rate has been chosen and why.
Projecting risk adjustments
Companies should also try to adjust the risk in the above calculations based on the specific project they are about to invest in. Unfortunately, the survey also found that many companies don’t
currently include risk adjustments in their cost of capital analysis, but rather just use a percentage point or more to the rate.
But this sort of standardization of cost of capital analysis can leave companies open to issues of overinvesting, for example. If you are calculating the cost of capital for a new investment project,
it is essential to also adjust the risks according to the project in question. This is especially important if the risk profile of the project varies greatly from the company’s own risk profile.
Limitations of WACC
Finally, you also need to keep in mind the limitations of WACC. It is crucial to remember the elements used in the formula are not consistent. These subtle differences can be apparent in the basic
calculations of how the company calculates its debt as well as its equity.
The final ratio you receive with WACC should therefore not be taken as the ultimate truth. Instead, you want to use the cost of capital as an important indicator, but also add other financial metrics
to your analysis and decision-making process. This is also an important point to remember if you are considering investing in a company.
The more you know about the financial status of the company to better. While the cost of capital needs to be taken with a pinch of salt and tough analysis, it is nonetheless an essential metric to
learn about.
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Allowing for catching-up ... a specific case
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I'm currently working on a piecepack-based design (link for piecepack info) that inflicted itself upon me the other day.
It's a coin-collecting game where each of the four players is a sorceror who is out to get as many spells (coins) as possible, by picking up the coins from the board and/or by winning them in 1-on-1
battles with the other sorcerors. The coins that a sorceror owns can be kept in 1 of two groups: (1) the stockpile is kept back at the sorceror's home and impacts the number of dice the sorceror can
roll during his turn (more coins = more dice); (2) the "wears" are kept with the sorceror and determine his attack and defense power in battles.
Now, since the numbers rolled on the dice can be used to move the sorceror around the board, it seems that a couple of great rolls will give one player a distinct advantage, because they will be able
to pick up more coins. So, to even things out in this regard (and to encourage battles "against the odds"), I've come up with the following scheme:
After a battle occurs (and the winner has been determined), the two sorcerors compare their total number of coins (stockpile + wears). If the winner has the same or more coins than the loser, they
can take a single coin (winner's choice) from the loser. If, however, the winner has fewer coins than the loser, they can take either all-but-one coin from the loser's wears, or all-but-one coin from
the loser's stockpile. (The loser is then allowed to re-distribute their remaining coins between the two groups.)
How well do you think this system would work? What problems will it lead to? I'm looking for the usual critical and clever opinions available at this forum. ;)
p.s. I realize that I haven't given you too much to go on, since I don't have a full set of the rules written-up and available for you yet. Just keep in mind that the piecepack itself limits the
number of available coins (24 total), that the initial board setup puts 1 or 2 coins on every square on the board, and that the main point of the game is for each sorceror to venture out from their
home, gather coins and battle as they see fit -- then return home to end the game (first sorcerer to get to their specific "touch square" and back home ends the game). Scoring will include points for
different kinds of coins collected (there are 4 different suits in the piecepack), best score in a suit, bonus points for being the only one to own a suit, and special scoring for the suit that
matches the player's color.
Wed, 08/27/2003 - 17:05
Re: Allowing for catching-up ... a specific case
Brykovian wrote:
After a battle occurs (and the winner has been determined), the two sorcerors compare their total number of coins (stockpile + wears). If the winner has the same or more coins than the loser,
they can take a single coin (winner's choice) from the loser. If, however, the winner has fewer coins than the loser, they can take either all-but-one coin from the loser's wears, or all-but-one
coin from the loser's stockpile. (The loser is then allowed to re-distribute their remaining coins between the two groups.)
How well do you think this system would work? What problems will it lead to? I'm looking for the usual critical and clever opinions available at this forum. ;)
It's difficult to say without knowing how many coins players will have with them, what they do, how combat is resolveds, how many coins are likely to be in peoples' caches at home, and why anyone
would fight if they can see that they won't win (I would assume therefore that "which coins are where" is hidden information).
It sounds to me like you are looking for a way to make a player move less each turn. This involves beating them in a fight so you can redistribute his coins (and take a bunch of them if they were
beating you, reversing the problem).
Frankly, I think that there must be a better way to balance out high movement than that. Off the top of my head I'd say it'd be better to limit the number of coins collected in a turn (like you move
any amount up to your movement, stopping at any time to pick up a coin on that space. Or maybe don't stop the movement, but you only get to pick up one coin or something.) I'm not saying that this
would be GOOD per se, I'm just saying it'd be better than concocting a crazy rule that's difficult to interperet and understand.
Your proposed adjustment is the very definition of the term "fiddley." Sounds very contrived and I'm not sure it even does what you are looking for.
Now that I think of it, the non-leader taking the leader's coins means they'll probably win the next fight too, which means not only are they the leader now, but they're a runaway leader.
- Seth
Wed, 08/27/2003 - 17:41
Allowing for catching-up ... a specific case
I'd say that instead of working on the combat system, you may want to limit the amount of coins available on the board. Instead of them being on every space, perhaps they can be in difficult-to-reach
corners of the board, so wizards can only get coins by a) schlepping, or b) beating up on the schleppers. b) should not be easy.
Some more ideas...
- Have an NPC guarding a single powerful coin. Defeating the NPC lets you get the treasure. Wizards can team up with each other to defeat the NPC guard, but only one gets the treasure! This opens up
an element of alliance and betrayal.
- At the beginning of the game, give each wizard a secret alignment. At the end of the game, the number of other players helped gets a Good wizard a bonus, while the number of other players defeated
gets an Evil wizard a bonus.
I hope this helps...
Wed, 08/27/2003 - 18:01
Allowing for catching-up ... a specific case
Base movement on coin encumberance: the more coins you're carrying, the fewer spaces you can move.
Wed, 08/27/2003 - 18:59
Allowing for catching-up ... a specific case
Brykovian wrote:
Thanks Seth ... you are right in that the stuff you don't know about the game may be more important than the stuff I was able to tell you about it. So ... I think I'll give you a bit more details
here. (Hopefully this will help me in putting together the official rules doc!)
Ahh, this makes more sense. If you pardon the analogy, it sounds like Goldland meets Tigris & Euphrates. :)
Perhaps I can rip off a bit of T&E for this suggestion: for a battle, both players pick up all coins in their storehouse, and secretly divide them into their hands. One hand (let's say the right
hand) contains extra coins they're willing to include directly from the storehouse for this battle, and the left hand holds the rest of the coins from the storehouse.
Both players make their selections and hold their right fists in front of them. They reveal simultaneously. All of these coins are factored into the battle, along with the coins from the
participants' wears.
Now, for the part you were worried about: the spoils...
The player who wins the battle looks at the coins in loser's right hand, and the coins in the loser's wears. He chooses a color, and the loser must give the winner all the coins but one of that color
across his wears and his hand. All other coins in the loser's hand return to the loser's storehouse. All coins in the winner's hand, plus the coins he won, go to the winner's storehouse. If the
winner selects a color that the loser has both in his hand and in his wears, the loser chooses whether the lone remaining coin will stay in his wears or return to his storehouse.
Obviously, I have no idea whether this system will work or not. You may have to cut down the number of coins the winner takes from the loser's wears, or the maximum number of coins a player can bid
(counting those both in his wears and in his hand). However, it certainly discourages a player with a lot of coins from attacking a player with a few coins. In fact, a player with only four coins
(one of each color) in his wears who bids nothing pays nothing!
One last suggestion: if you implement this system, you may want to change the movement system so that each coin in a player's storehouse of a different color earns him an extra die. Yes, another T&
E-influenced suggestion; sorry, I can't help it, argh...
Hope this helps!
Wed, 08/27/2003 - 19:29
Allowing for catching-up ... a specific case
IngredientX wrote:
One last suggestion: if you implement this system, you may want to change the movement system so that each coin in a player's storehouse of a different color earns him an extra die.
I don't know about the combat system Gil mentioned, but this sounds like a groovy idea- you can still get 1-4 dice per turn, but it's a lot harder to get all 4. Have players start with one of their
own coins in their stock, so you don't have to worry about what happens when someone has no coins (i.e. do they get no dice?)
- Seth
Wed, 08/27/2003 - 20:06
Allowing for catching-up ... a specific case
Battles are determined by comparing each player's wears. For each sorceror, coins in their matching color represent their defense. Coins in their opponent's matching color represent their attack.
Take each player's defense minus their opponent's attack, and the highest result wins. There are a few things to make these battles a bit more interesting:
"Null" coins (piecepack coins are numbered: Null,Ace,2,3,4,5) can be used to "block" any of the opponent's coins of matching color
Coins of the other 2 colors are used if a tie-breaker is needed
This sounds really cool (I would buy a piecepack set to play this if all the kinks were worked out). Collect coins to increase offence / defence. And while I see some similarities with T&E, I don't
think that it's near enough to draw too many comparisons.
Players can arrange, stack, display their stockpile and wears coins in any way they wish ... so it may not be easy to tell exactly what a player has
This is the only part I don't like. To me it seems that it should either be mandatory for players to arrange their piles so that all can see what they have, or it should be completely hidden. This
just feels too fiddly.
Wed, 08/27/2003 - 20:42
Allowing for catching-up ... a specific case
Oh yeah, and you should have to carry around the coins you pick up, unless you want to stop off at home and drop them off. The game end could be signalled when a particular coin is dropped off at a
player's home (the far one).
- Seth
Thu, 08/28/2003 - 01:45
Allowing for catching-up ... a specific case
Brykovian wrote:
I like Gil's dice-for-different colors idea as well. (btw, players always get to shake at least 1 die, even if they have no coins in the stockpile.)
Yes, I read that. My point was if you just start with your own colored coin in your stock, then the rule is simply "Roll 1 die per type of coin each turn" instead of "Roll 1 die per type of coin each
turn. If you have no coins, you roll 1 die."
Also, maybe consider this- rather than dice, just move up to X spaces, where X is the number of coin types you have in stock.
Seth, you recommend starting them with a coin of their own color -- I suppose the 3 coin would be the best option ... kinda middle-of-the-road. But, it's probably not necessary since you'll
always have at least 1 die to roll.
I think perhaps I missed something. Does it make a difference which coin(s) you have in stock? I thought it was just the number of coins (originally), or the number of types (more recently).
I could also see having everyone move the same number of squares on the first turn (moving 3 squares would get everyone 4 coins on that first turn, if they wanted). Then we're just down to the
random distribution of the coins that'll impact the starting situation.
This sounds terrible to me. Seems like there should be some decision or some trade-off to getting to move farther each turn. If everyone just gets to ramp up to max speed right off the bat, then
what's the point? Everyone's going to do it as a matter of course- not just because they want to, but because they're supposed to be getting coins anyway.
I MUST have missed SOMETHING, because I don't see any decisions to make in this game at all. You move to the center and back, collecting as many coins as possible en route. If you're lucky (or plan a
little) you might happen upon another sorcerer who's not as strong at the moment as you are, and you can beat them up and take a coin. Or, if they're currently ahead of you you can beat them up and
take MOST of their coins, ensuring they won't come back after you right away, then you can run home to victory.
sej wrote:
...carry around the coins you pick up, unless you want to stop off at home and drop them off. The game end could be signalled when a particular coin is dropped off at a player's home (the far
This would add a new dimension, but add to the length of the game. Players could drag newly-found coins around with them (perhaps, mechanically, under their pawn) until they were adjacent to
their home -- then they'd go in their stockpile. This would probably lead to some players making short trips out into the world and back home again ... not sure if I like it, but something worth
keeping in mind, and definitely an optional/variant rule.
So you've introduced a decision... do I make short trips and then stop off at home (which could be seperated from the coins so it's a real schlep)? Or do I collect a lot of coins- which ought to be
risky- maybe you can lose them that way (If someone beats you up, you lose some or all of your "wear" coins). To round it out, you could have a maximum carrying capacity... and/or have your maximum
movement determined by the stockpile, but then -1 square per coin you're carrying. Something like that.
You need something to drive the player's decisions. "Do I do X, or do I do Y instead?" Not "Of course I collect coins and only attack if I'll win, because there's no other option."
But, back to my original reason for posting ... assuming that some players will "make a haul" off a good start, and other players fall behind ... would having the 2-level battle spoils system
help correct for that? I don't think the idea is overly-fiddly -- just gives an extra reason to attack someone in a better position than you.
I still don't see how it will help and only see how it could hurt.
- Seth
Thu, 08/28/2003 - 01:58
Allowing for catching-up ... a specific case
Brykovian wrote:
I like Gil's dice-for-different colors idea as well. (btw, players always get to shake at least 1 die, even if they have no coins in the stockpile.)
Yes, I read that. My point was if you just start with your own colored coin in your stock, then the rule is simply "Roll 1 die per type of coin each turn" instead of "Roll 1 die per type of coin each
turn. If you have no coins, you roll 1 die."
Also, maybe consider this- rather than dice, just move up to X spaces, where X is the number of coin types you have in stock.
Seth, you recommend starting them with a coin of their own color -- I suppose the 3 coin would be the best option ... kinda middle-of-the-road. But, it's probably not necessary since you'll
always have at least 1 die to roll.
I think perhaps I missed something. Does it make a difference which coin(s) you have in stock? I thought it was just the number of coins (originally), or the number of types (more recently).
I could also see having everyone move the same number of squares on the first turn (moving 3 squares would get everyone 4 coins on that first turn, if they wanted). Then we're just down to the
random distribution of the coins that'll impact the starting situation.
This sounds terrible to me. Seems like there should be some decision or some trade-off to getting to move farther each turn. If everyone just gets to ramp up to max speed right off the bat, then
what's the point? Everyone's going to do it as a matter of course- not just because they want to, but because they're supposed to be getting coins anyway.
I MUST have missed SOMETHING, because I don't see any decisions to make in this game at all. You move to the center and back, collecting as many coins as possible en route. If you're lucky (or plan a
little) you might happen upon another sorcerer who's not as strong at the moment as you are, and you can beat them up and take a coin. Or, if they're currently ahead of you you can beat them up and
take MOST of their coins, ensuring they won't come back after you right away, then you can run home to victory.
sej wrote:
...carry around the coins you pick up, unless you want to stop off at home and drop them off. The game end could be signalled when a particular coin is dropped off at a player's home (the far
This would add a new dimension, but add to the length of the game. Players could drag newly-found coins around with them (perhaps, mechanically, under their pawn) until they were adjacent to
their home -- then they'd go in their stockpile. This would probably lead to some players making short trips out into the world and back home again ... not sure if I like it, but something worth
keeping in mind, and definitely an optional/variant rule.
So you've introduced a decision... do I make short trips and then stop off at home (which could be seperated from the coins so it's a real schlep)? Or do I collect a lot of coins- which ought to be
risky- maybe you can lose them that way (If someone beats you up, you lose some or all of your "wear" coins). To round it out, you could have a maximum carrying capacity... and/or have your maximum
movement determined by the stockpile, but then -1 square per coin you're carrying. Something like that.
You need something to drive the player's decisions. "Do I do X, or do I do Y instead?" Not "Of course I collect coins and only attack if I'll win, because there's no other option."
But, back to my original reason for posting ... assuming that some players will "make a haul" off a good start, and other players fall behind ... would having the 2-level battle spoils system
help correct for that? I don't think the idea is overly-fiddly -- just gives an extra reason to attack someone in a better position than you.
I still don't see how it will help and only see how it could hurt.
- Seth
Thu, 08/28/2003 - 08:55
Allowing for catching-up ... a specific case
Yeah ... I think I wasn't clear enough on some points. The number of different suits of the coins in the stockpile (I've already incorporated Gil's suggestion ;))
Kewl, thanx!
This game *is* a lot simpler than a lot of games discuss at these forums (the references to T&E or Goldland kinda go over my head since I've never played them, and trying to read their rules have
left me confused :() ... but I think it's simply not true that there aren't *any* decisions to be made.
Goldland is a very good schlepping game, in which players lay down tiles to create a wilderness they are exploring. They pick up goods during the game and hold them in their inventory. The amount of
stuff they have in their inventory determines the distance they can move; players with a smaller inventory can move further. This is a mechanism you may want to look at; you might even be able to
eliminate dice altogether, with the different suits of coins in a player's storehouse determining the number of actions he may make in a turn, and the amount of wears he his carrying negatively
impacting his movement ability.
T&E's scoring system has drawn a lot of praise. During the game, you score four differently-colored tiles. Your final score is the color you have the LEAST of. This forces you to diversify. This
would be a great scoring system for you to (ahem) borrow, with players only scoring the suit they have the least of. It de-emphasizes the number of total coins in the storehouse, and keeps players
from trying to hoard a single suit.
Back to the spoils-of-battles for sec ... Since the different suits being held are more important, I'm thinking of doing the following:
□ Players stack same-suited coins together and display them suit-side-up in both their stockpile and their wears
□ If the winner of a battle has the same number or more total coins (stockpile + wears) than the loser, then the winner can chose any suit stack of coins from the loser's wears.
□ If the winner of a battle has fewer total coins than the loser, then the winner can chose any suit, and get all coins of that suit from the loser's wears *and* stockpile.
I'm still not crazy about this, because it might skew a little too much towards bash-the-leader. Of course, I haven't actually played it out, so no one will know for sure until then...
What I was trying to go for earlier was a self-balancing mechanism, where a player with a lot of coins is naturally more liable to give up lots of coins than a player with very few coins. IMVHO, the
storehouse should be completely "safe," with no chance of another player stealing from it in battle. With rules (ahem ahem) "liberated" from T&E, the raw amount of coins you have in the storehouse is
irrelevant; if you have no coins of at least one color, your score is zero!
But I think you should shoot for a battle mechanic where a rich player is, by nature, more vulnerable than a poor player. Now that I think of it, I'll amend my earlier battle suggestion so that the
winner of a battle takes ALL coins of the loser's suit in his wears, plus ALL coins of the same suit that he decided to risk from his storehouse.
Perhaps you can figure out a nasty mechanism that has nothing to do with battle, that lets one player plunder another player's storehouse...
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esponse of
Time response of sampled-data feedback system
[vt,yt,ut,t] = sdlsim(p,k,w,t,tf)
[vt,yt,ut,t] = sdlsim(p,k,w,t,tf,x0,z0,int)
sdlsim(p,k,w,t,tf) plots the time response of the hybrid feedback system. lft(p,k), is forced by the continuous input signal described by w and t (values and times, as in lsim). p must be a
continuous-time LTI system, and k must be discrete-time LTI system with a specified sample time (the unspecified sample time –1 is not allowed). The final time is specified with tf.
sdlsim(p,k,w,t,tf,x0,z0) specifies the initial state vector x0 of p, and z0 of k, at time t(1).
sdlsim(p,k,w,t,tf,x0,z0,int) specifies the continuous-time integration step size int. sdlsim forces int = (k.Ts)/N int where N>4 is an integer. If any of these optional arguments is omitted, or
passed as empty matrices, then default values are used. The default value for x0 and z0 is zero. Nonzero initial conditions are allowed for p (and/or k) only if p (and/or k) is an ss object.
If p and/or k is an LTI array with consistent array dimensions, then the time simulation is performed pointwise across the array dimensions.
[vt,yt,ut,t] = sdlsim(p,k,w,t,tf) computes the continuous-time response of the hybrid feedback system lft(p,k) forced by the continuous input signal defined by w and t (values and times, as in lsim).
p must be a continuous-time system, and k must be discrete-time, with a specified sample time (the unspecified sample time –1 is not allowed). The final time is specified with tf. The outputs vt, yt
and ut are 2-by-1 cell arrays: in each the first entry is a time vector, and the second entry is the signal values. Stored in this manner, the signal vt is plotted by using one of the following
Signals yt and ut are respectively the input to k and output of k.
If p and/or k are LTI arrays with consistent array dimensions, then the time simulation is performed pointwise across the array dimensions. The outputs are 2-by-1-by-array dimension cell arrays. All
responses can be plotted simultaneously, for example, plot(vt).
[vt,yt,ut,t] = sdlsim(p,k,w,t,tf,x0,z0,int) The optional arguments are int (integration step size), x0 (initial condition for p), and z0 (initial condition for k). sdlsim forces int = (k.Ts)/N, where
N>4 is an integer. If any of these arguments is omitted, or passed as empty matrices, then default values are used. The default value for x0 and z0 is zero. Nonzero initial conditions are allowed for
p (and/or k) only if p (and/or k) is an ss object.
Time Response of Continuous Plant with Discrete Controller
To illustrate the use of sdlsim, consider the application of a discrete controller to a plant with an integrator and near integrator. A continuous plant and a discrete controller are created. A
sample-and-hold equivalent of the plant is formed and the discrete closed-loop system is calculated. Simulating this gives the system response at the sample points. sdlsim is then used to calculate
the intersample behavior.
P = tf(1,[1, 1e-5,0]);
T = 1.0/20;
C = ss([-1.5 T/4; -2/T -.5],[ .5 2;1/T 1/T],...
[-1/T^2 -1.5/T], [1/T^2 0],T);
Pd = c2d(P,T,'zoh');
Use connect to construct the interconnected feedback system.
C.InputName = {'ref','y'};
C.OutputName = 'u';
Pd.Inputname = 'u';
Pd.OutputName = 'y';
dclp = connect(C,Pd,'ref','y');
Use step to simulate the digital step response.
[yd,td] = step(dclp,20*T);
Set up the continuous interconnection and calculate the sampled data response with sdlsim.
M = [0,1;1,0;0,1]*blkdiag(1,P);
t = [0:.01:1]';
u = ones(size(t));
y1 = sdlsim(M,C,u,t);
xlabel('Time: seconds')
title('Step response: discrete (*) and continuous')
You can see the effect of a nonzero initial condition in the continuous-time system. Note how examining the system at only the sample points will underestimate the amplitude of the overshoot.
y2 = sdlsim(M,C,u,t,1,0,[0.25;0]);
xlabel('Time: seconds')
title('Step response: nonzero initial condition')
Finally, you can examine the effect of a sinusoidal disturbance at the continuous-time plant output. This controller is not designed to reject such a disturbance and the system does not contain
antialiasing filters. Simulating the effect of antialiasing filters is easily accomplished by including them in the continuous interconnection structure.
M2 = [0,1,1;1,0,0;0,1,1]*blkdiag(1,1,P);
t = [0:.001:1]';
dist = 0.1*sin(41*t);
u = ones(size(t));
[y3,meas,act] = sdlsim(M2,C,[u dist],t,1);
xlabel('Time: seconds')
title('Step response: disturbance (dashed) and output (solid)')
sdlsim oversamples the continuous-time, N times the sample rate of the controller k.
Version History
Introduced before R2006a | {"url":"https://www.mathworks.com/help/robust/ref/dynamicsystem.sdlsim.html","timestamp":"2024-11-07T20:07:56Z","content_type":"text/html","content_length":"83557","record_id":"<urn:uuid:9af595f0-d3d3-409c-97a6-9461e3c50df3>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00144.warc.gz"} |
numpy.take(a, indices, axis=None, out=None, mode='raise')[source]¶
Take elements from an array along an axis.
This function does the same thing as “fancy” indexing (indexing arrays using arrays); however, it can be easier to use if you need elements along a given axis.
a : array_like
The source array.
indices : array_like
The indices of the values to extract.
New in version 1.8.0.
Also allow scalars for indices.
axis : int, optional
The axis over which to select values. By default, the flattened input array is used.
out : ndarray, optional
If provided, the result will be placed in this array. It should be of the appropriate shape and dtype.
mode : {‘raise’, ‘wrap’, ‘clip’}, optional
Specifies how out-of-bounds indices will behave.
□ ‘raise’ – raise an error (default)
□ ‘wrap’ – wrap around
□ ‘clip’ – clip to the range
‘clip’ mode means that all indices that are too large are replaced by the index that addresses the last element along that axis. Note that this disables indexing with negative
subarray : ndarray
The returned array has the same type as a.
See also
Take elements using a boolean mask
equivalent method
>>> a = [4, 3, 5, 7, 6, 8]
>>> indices = [0, 1, 4]
>>> np.take(a, indices)
array([4, 3, 6])
In this example if a is an ndarray, “fancy” indexing can be used.
>>> a = np.array(a)
>>> a[indices]
array([4, 3, 6])
If indices is not one dimensional, the output also has these dimensions.
>>> np.take(a, [[0, 1], [2, 3]])
array([[4, 3],
[5, 7]]) | {"url":"https://devdoc.net/python/numpy-1.12.0/reference/generated/numpy.take.html","timestamp":"2024-11-09T20:31:37Z","content_type":"text/html","content_length":"10118","record_id":"<urn:uuid:61ba95b5-4fda-4d44-8b8e-f148cc853acf>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00372.warc.gz"} |
Slope Calculator – BizCalcs.com
Slope Calculator
Our dedicated slope calculator will help you find the slope (m) or the gradient between two points. All you have to do is follow the instructions mentioned on the calculator!
Enter Information
Fill the calculator form and click on Calculate button to get result here
Slope (m) 0
Angle (deg) 0
Understanding Slope and Its Significance
The slope of a line is the change in the y coordinate with respect to the change in the x coordinate.
m = change in y/change in x = Δy/Δx
• m is the slope
• net change in the y-coordinate is represented by Δy
• net change in the x-coordinate is represented by Δx
Additionally, the slope of a line can also be represented by: tan θ = Δy/Δx (tan θ is the slope of a line)
Different Types Of Slopes
There are four different types of slopes, which are detailed here:
Slope Type Description Formula Example
Positive Slope The line rises from left to right. \(m > 0\) \(y = 2x + 3\)
Negative Slope The line falls from left to right. \(m < 0\) \(y = -4x + 2\)
Zero Slope The line is horizontal. \(m = 0\) \(y = 5\)
Undefined Slope The line is vertical. The denominator of the slope is zero. \(x = 4\)
Overview Of Slope In Mathematical Analysis
A slope is like a ladder’s angle against a wall. It tells you how much a line tilts up or down in math. Think about climbing hills: A steep hill has a high slope, and a gentle hill has a low slope.
In maths, we use the slope to see how things change together. For example, if you know how fast you walk and the time it takes to reach somewhere, you can find out how far away your destination is.
The formula for finding the slope uses two points on a straight line. You take the difference in their y-coordinates and divide it by the difference in their x-coordinates. This gives you the rate of
change or slope of that line.
The steeper the line, the bigger this number gets. If lines go up from left to right they have positive slopes; if they go down they have negative slopes. Slopes help us predict things and solve
problems in fields like science and engineering where understanding rates of change is key!
Explanation Of Slope Formulas
Now that you know how important slope is, let’s talk about how we find it. Finding the slope of a line means figuring out how steep it is. Picture walking up a hill; the steeper the hill, the harder
you work to climb it.
In math, we measure this “steepness” with numbers.
Here’s one way to calculate slope: take two points on a line and look at their x and y values (these are just places on the graph). Subtract the y value of one point from the other and do the same
for x.
This gives us two numbers. We call these differences numerators and denominators. Then we divide them – numerators over denominators – which gives us what we call ‘the rise over run‘.
The bigger this number, the steeper your line.
If you have ever seen an equation like “y = mx + b”, that’s called slope-intercept form. Here, m represents our slope—the thing we talked about as being like climbing a hill—and b is where our line
starts off when x equals zero on our graph or what some people would call ‘crossing’ the y-axis.
Calculating slopes isn’t only for straight lines; sometimes curves in graphs go up and down, not just straight across – those are non-linear functions! And if someone throws words like hypotenuse or
Pythagorean theorem at you while talking about slopes—don’t worry! Those come into play when dealing with right triangles found in more advanced problems involving distance and angles on graphs.
Remember, knowing these formulas helps make sense of things around us everywhere—from ramps to mountainsides to even roller coaster tracks!
Step-By-Step Guide For Calculating The Equation Of A Line
Calculating the equation of a line can seem tough, but with this guide, you’ll learn how to do it step by step. We’ll use slope formulas and simple math to find the right line equation from two
• First, write down the coordinates of your two points (x1, y1) and (x2, y2).
• Next, subtract y2 from y1 to get the change in y or Δy.
• Subtract x2 from x1 to get the change in x or Δx.
• Use these changes to find the slope (m). Divide Δy by Δx (m = Δy / Δx).
• The slope shows how steep the line is. A bigger slope means a steeper line.
• With the slope (m), choose one point (x1, y1) for the next steps.
• Get ready to use the point-slope form: y – y1 = m(x – x1).
• Plug in your point’s values and m into this formula.
• Multiply m with (x – x1) to simplify things.
• Add y1 to move it over to the other side of your equation.
1. Take note of your two points
2. Calculate changes in y and x
3. Find your slope by dividing those changes
4. Use a single point with your slope in point-slope form
5. Simplify your formula
Features Of The Slope Calculator
The Slope Calculator empowers students to effortlessly conquer the complexities of slope calculations, offering a robust array of functionalities designed to handle various input data with precision.
From exploring different methods of computing slopes to delving into related concepts like y-intercept and angle measurements, this tool is engineered for comprehensive analytical support.
Input Parameters And Values
Slope calculators are helpful tools for math. They let you find the slope of a line easily. Here’s how you put information into a slope calculator:
• You can use whole numbers, fractions, or mixed numbers.
• For two points, enter the x and y coordinates like (2,3) and (4,5).
• If you know one point and the slope, just type them in.
• Want to use distance? Put in one point with the slope and how far it goes.
• To find out about x or y alone, input one point with the slope and either the x or y value.
• Just have a line’s equation? Enter it to get the slope.
Different Calculation Methods
Calculating slope is important in math. It helps us understand how steep a line is. Here are some ways you can figure out the slope with a calculator:
• Two Points: You need two points for this method. Each point has an “x” and a “y” value, like (x1, y1) and (x2, y2). The calculator finds the vertical change (ΔY) and horizontal change (ΔX)
between them. Then it uses the slope formula: (y2 – y1) / (x2 – x1). This gives you the slope of the line.
• One Point with Slope (m) & Distance: If you have one point and know the slope and how far another point is from it, use this method. Give your points “x” and “y”, type in the slope (m), and add
the distance. The calculator will show you where the second point is.
• One Point with Slope (m) & X or Y: Here, start with one point and the slope again. But this time, only add either a new “x” value or a new “y” value. This will help you find another point on that
same line.
• One Point & Slope (m): Have just one point? That’s okay! Type in its “x” and “y” values along with the slope (m). The calculator then makes an equation of a line in slope-intercept form: y = mx +
• A Line: Sometimes, you might already have an equation of a line like y = mx + b. Just put that into the calculator! It will tell you about the line’s steepness and where it crosses the y-axis.
Additional Options For Y-Intercept And Angle Calculation
The slope calculator is a handy tool for math students. It not only finds the slope but also calculates the y-intercept and angle of a line.
• Use two points on a line to find the slope. This method shows how steep the line is.
• Find out where your line crosses the y-axis. The calculator will give you this point, called the y-intercept.
• Learn about the angle your line makes with the x-axis. The calculator tells you this in degrees.
• Input one point and the slope (m). With these, you’ll see where your line goes and its steepness.
• Calculate the midpoint between two points using the midpoint calculator option.
• If you have a vertical line, know that it has an undefined slope. The calculator will tell you so because dividing by zero doesn’t work in math.
• For horizontal lines, get comfortable knowing they have a zero slope. It’s like a flat road without any hills.
How To Use The Slope Calculator
4. How to Use the Slope Calculator: Discovering the simplicity of slope calculations is just a few clicks away, as we guide you through inputting data and interpreting the precise results our tool
provides—empowering your mathematical journey with confidence and clarity.
Input Options And Output Results
A slope calculator helps you find the steepness of a line. It’s easy to use and gives you quick answers. Here’s what you can do with it:
• Choose how you want to find the slope. You can use two points, one point and the slope & distance, or other ways.
• Type in the numbers you know. These could be whole numbers, fractions, or mixed numbers.
• Press enter and see your results right away. The calculator tells you many things like:
• Slope (m): This is your main answer and shows how steep the line is.
• Percentage grade: It tells you the slope as a percentage.
• Angle (θ): This shares the tilt of the line in degrees.
• Distance: Find out how long the line is between points.
• ΔX and ΔY: These show how much x and y change between your two points.
• X-Intercept: Learn where the line crosses the x-axis.
• Y-Intercept: See where it crosses the y-axis on your graph.
Examples For Each Calculation Method
Slope calculators are handy tools for students learning about lines and angles. They make finding slopes quick and easy. Here are some ways to use different methods on the slope calculator:
1. Enter the x and y coordinates of two points.
2. The calculator will show the slope (m), distance, ΔX (change in x), ΔY (change in y), and the equation of the line.
• One Point with Slope (m) & Distance:
1. Type in one point’s x and y values.
2. Add the slope value (m) and the distance from the point.
3. Results will include a second point that creates this slope at that distance.
• One Point with Slope (m) & X or Y:
1. Put in an x or y coordinate along with the slope value (m).
2. The calculator figures out the other point needed to achieve this slope.
1. Input one point’s coordinates.
2. Provide the value for slope (m).
3. Get results for what line passes through this point at this angle, including its equation.
1. Give any line’s equation.
2. Find out its rise over run, which is another way to say its slope.
Question: How Does A Slope Calculator Use Point-Slope Form?
The slope calculator uses point slope form to create an equation from one point and the slope, helping you understand how lines go up or down on a graph.
Question: Can I Use The Pythagorean Theorem With A Slope Calculator?
Yes, you can use the Pythagorean theorem in some cases with your slop calculations to find distances when working with right triangles and altitude.
Question: What Happens If I Try Dividing By Zero In A Slope Calculation?
In math, division by zero isn’t allowed because it doesn’t make sense—it means you’re trying to split something into zero parts! A good rule is to remember: that we can’t divide by zero.
Question: Does Differential Calculus Relate To Finding Slopes Of Curves Rather Than Straight Lines?
Yes, differential calculus involves differentiating polynomials and other functions to find out things like curvature—how curvy or bendy—a curve is instead of just looking at straight lines.
Related Mathematics Calculators
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Re: st: trying to combine local macro and "format" command in a loop
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Re: st: trying to combine local macro and "format" command in a loop
From Nick Cox <[email protected]>
To [email protected]
Subject Re: st: trying to combine local macro and "format" command in a loop
Date Thu, 5 May 2011 19:00:44 +0100
The program is called Stata, not STATA.
r(mean) and r(sd) are numbers, not strings. For the string
interpretation, you want say
but the code segment
local lm = length(r(mean)); local ls = length(r(sd));
local mn`v'_snh`i': display %`lm'.2f r(mean);
local std`v'_snh`i': display %`ls'.2f r(sd);
still looks unnecessarily contorted -- and not even likely to produce
nice displays. Something like
local show = trim("`: display %10.2f r(mean)'")
shows a more useful approach.
It sounds as if you want 2 decimal places. So use a format such as
%10.2f that will always be big enough, then use -trim()- to get rid of
unwanted spaces. (Your value of "10" may be different.)
On Thu, May 5, 2011 at 5:57 PM, Woolton Lee <[email protected]> wrote:
> Thank you to Maarten Buis for help in bringing some resolution to my
> ongoing problem. I've modified my code so that it now attempts to
> format the local macros created so that
> 1) they are rounded to two decimals
> 2) the assigned format matches the length of the string stored in a macro
> These local macros are used in a postfile command to store the results
> in a table. I'm using the following code to set the format of each
> macro so that it matches the actual length of the numeric value. The
> variables included vary in length for example age is like 52.34, while
> total charges (tchg) can be something like 539202.12. I use the
> function length to set a local macro for the mean and for the length
> of the standard deviation of each variable and then attempt to set the
> format.
> /* continuous variables - Lung */
> local vars2 age los tchg costpd rbchg rcchg scchg aneschg phrchg radchg mrict
> nmchg clchg orchg msschg othchg;
> forvalues x = 1/16 {;
> local v: word `x' of `vars2';
> /********* insurance by safety net hospital */
> forvalues a = 1/4 {;
> local i: word `a' of `ins';
> sum `v' if vhi_site == "Lung" & `i' == 1 & snh == 1 & link_lung == 1;
> local lm = length(r(mean)); local ls = length(r(sd));
> local mn`v'_snh`i': display %`lm'.2f r(mean);
> local std`v'_snh`i': display %`ls'.2f r(sd);
> };
> However, STATA gives me the following error.
> type mismatch
> Any ideas how to fix this problem so that in the loop each macro is
> assigned its ROUNDED length?
> From maarten buis <[email protected]>
> To [email protected]
> On Thu, May 5, 2011 at 5:44 PM, Woolton Lee <[email protected]> wrote:
>> I have a program which creates descriptive statistics using tab1,
>> summarize and other functions, stores the macros then posts them using
>> postfile and creates tables that can be cut and pasted easily into a
>> word document.
> <snip>
>> This loop creates the variables mn and sd and rounds the result of the
>> numbers I want to two decimal places then uses summarize to store
>> these values into macros. I wonder if there is another more efficient
>> way to do this?
> Yes, use the extended macro function -: display-, like in the example
> below:
> *-------------- begin example -----------------------
> sysuse auto, clear
> ds , has(type numeric)
> local vars2 `r(varlist)'
> foreach v of local vars2 {
> sum `v'
> local m`v' : display %9.2f r(mean)
> local sd`v' : display %9.2f r(sd)
> }
> foreach v of local vars2 {
> di as txt "mean of `v' = " as result `m`v''
> }
> *-------------------- end example ----------------------
> You can read more about extended macro functions by typing in
> Stata -help extended_fcn- (I never remember the exact name, so
> I always type -help macro- or -help local- and than click on the link
> for extended macro functions).
* For searches and help try:
* http://www.stata.com/help.cgi?search
* http://www.stata.com/support/statalist/faq
* http://www.ats.ucla.edu/stat/stata/ | {"url":"https://www.stata.com/statalist/archive/2011-05/msg00269.html","timestamp":"2024-11-15T02:37:06Z","content_type":"text/html","content_length":"14145","record_id":"<urn:uuid:4e5bc200-c54a-4d9d-8df7-f9d8ff046ee0>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00223.warc.gz"} |
Program of
Mathematical Methods (part B)
(Prof. M. Ferri)
The course consists of two parts. Part A is held by Prof. Nicola Arcozzi Ph.D. Part B is dedicated to Graph Theory.
The following program will be updated during the course.
1. Graphs and subgraphs (1.1 to 1.7; 1.8 up to first 24 lines of page 19).
Theorems: Thm. 1.1 (with proof), Cor. 1.1 (with proof), Thm.1.2 (with proof)
Exercises: 1.1.3, 1.2.1, 1.2.3, 1.2.4, 1.2.10, 1.2.12(c), 1.3.1, 1.4.4, 1.5.1, 1.5.4, 1.5.5, 1.5.6(a), 1.6.1, 1.6.11, 1.7.2.
2. Trees (2.1; 2.2 up to Cor. 2.4.2; 2.3 up to Thm. 2.7; 2.4; 2.5 up to Thm. 2.10 excluded).
Theorems: Thm. 2.1 (with proof), Thm.2.2 (with proof), Cor. 2.2, Thm. 2.3, Thm. 2.4, Cor. 2.4.1, Cor. 2.4.2 (with proof), Thm. 2.7, Thm. 2.8 (with proof), Thm. 2.9.
Exercises: 2.4.3.
3. Connectivity (3.1; 3.2 up to Cor. 3.2.2).
Theorems: Thm. 3.1, Thm. 3.2, Cor. 3.2.1, Cor. 3.2.2
4. Euler tours and Hamilton cycles (4.1; 4.2 up to Cor. 4.4; 4.3 up to Thm. 4.7; 4.4: first 10 lines).
Theorems: Thm. 4.1, Cor. 4.1, Thm. 4.2, Thm. 4.3 (with proof), Thm. 4.4, Cor. 4.4, Thm. 4.7.
Exercises: 4.1.1, 4.1.2.
5. Matchings (5.1, 5.2; 5.4: first 8 lines).
Theorems: Thm. 5.1, Thm. 5.2, Cor. 5.2, Lemma 5.3, Thm. 5.3.
Exercises: 5.1.1(a), 5.1.5(b).
6. Edge colourings (6.1: first 20 lines, Thm. 6.1; 6.2: Thm. 6.2; 6.3: first 21 lines).
Theorems: Thm. 6.1, Thm. 6.2.
7. Independent sets and cliques (7.1; 7.2 up to the table of numbers).
Theorems: Thm. 7.1 (with proof), Cor. 7.1 (with proof), Thm. 7.2, Thm. 7.3, Thm. 7.4.
8. Vertex colourings (8.1 up to Cor. 8.1.2; 8.2; 8.3: only Thm. 8.5; 8.4; 8.5 up to Thm. 8.7; 8.6).
Theorems: Thm. 8.1 (with proof), Cor. 8.1.1 (with proof), Cor. 8.1.2 (with proof), Thm. 8.4, Thm. 8.5, Thm. 8.6 (with proof), Cor. 8.6, Thm. 8.7.
Exercises: 8.4.1, 8.4.2, 8.4.3(a) 8.6.2.
9. Planar graphs (9.1 excluding Thm. 9.1; 9.2; 9.3; 9.5 without lemmas and without proof; 9.6 up to Thm. 9.12 excluded, without proof; mind the footnotes!).
Theorems: Thm. 9.2 (with proof), Thm. 9.3 (with proof), Thm. 9.4 (with proof), Thm. 9.5 (with proof), Cor. 9.5.1 (with proof), Cor. 9.5.2 (with proof), Cor. 9.5.3 (with proof), Cor. 9.5.4 (with
proof), Cor. 9.5.5 (with proof), Thm. 9.10, Thm. 9.11, footnote at p. 157.
Exercises: 9.1.2.
10. Directed graphs (10.1; 10.2 up to Cor. 10.1; 10.4 first 17 lines; 10.5; 10.6 up to Thm. 10.5).
Theorems: Thm. 10.1 (with proof), Cor. 10.1 (with proof), Thm. 10.5 (with proof).
Exercises: 10.1.1
There will be short expositions of basic Topological Data Analysis (NOT part of the exam program): Gradient, divergence and Laplacian on graphs , the "smallworld" model for networks, some persistent
Modelling problems with graphs. Writing matrices associated with graphs. Computing graph invariants. Solving elementary graph problems: Shortest path problem (by Dijkstra's Algorithm), Connector
problem (by Kruskal's algorithm), Chinese postman problem (by Fleury's algorithm), finding minimal coverings and maximal independent sets (by logical operations), making a 2-edge-connected graph
diconnected. Building a maximal flow in a network.
The student is bound to (try to) solve the book exercises listed above for each chapter.
Official textbook:
J.A. Bondy and U.S.R. Murty, "Graph theory with applications",
North Holland, 1976.
Freely downloadable at https://www.zib.de/groetschel/teaching/WS1314/BondyMurtyGTWA.pdf (24 MB).
Support textbooks:
J.A. Bondy and U.S.R. Murty, "Graph theory",
Springer Series: Graduate Texts in Mathematics, Vol. 244 (2008)
R. Diestel, "Graph theory", Springer Series: Graduate Texts in Mathematics, Vol. 173 (2005)
Freely downloadable at http://diestel-graph-theory.com/basic.html (3 MB).
A mid-term test will be given on December 13, 2024, during class; no booking needed. The mid-term test MUST be passed with a score of at least 14 (over 24). If you don't pass, you must recover it.
Apply for the final exam at AlmaEsami. Booking is possible only after the end of the course and up to two days before the exam. The final exam is on the whole program above and is as follows: I
propose two subjects (each of which is either the title of a long chapter, or the sum of the titles of two short ones); you choose one and write down all what you remember about it, and then we
discuss on your essay and in general about the chosen subject. It is an oral examination, so writing is only a help for you to gather ideas.
ATTENTION! If you need to be examined in a precise day of the proposed ones, please register ahead of time: if the maximum number (30) of students for a shift has been reached, the system
automatically registers you at the following shift!
Supplementary material:
Example 1 of mid-term test and its solution.
Example 2 of mid-term test and its solution.
Have a look at the demos of graph theory.
Recordings of year 2024
Recording and whiteboard of the lecture of October 25, 2024.
Recording and whiteboard of the lecture of October 18, 2024.
Recording and slides of the lecture of October 11, 2024.
First part, second part and whiteboard of the lecture of September 27, 2024.
First part, second part and whiteboard of the lecture of September 20, 2024.
Recordings of year 2023
First part, second part and whiteboard of the lecture of December 19, 2023.
Recording and whiteboard of the lecture of December 12, 2023.
Recording and whiteboard of the lecture of December 5, 2023.
Recording and whiteboard of the lecture of December 1, 2023.
Recording and whiteboard of the lecture of November 28, 2023.
Recording and whiteboard of the lecture of November 24, 2023.
Recording and whiteboard of the lecture of November 21, 2023.
Recording and whiteboard of the lecture of November 17, 2023.
Recording and whiteboard of the lecture of November 10, 2023.
Recording and whiteboard of the lecture of October 27, 2023.
Recording and whiteboard of the lecture of October 24, 2023.
Recording and whiteboard of the lecture of October 13, 2023.
Recordings of year 2022
Recording of the seminars of December 20, 2022.
Recording and whiteboard of the lecture of December 16, 2022.
First part, second part and whiteboard of the lecture of December 13, 2022.
Recording and whiteboard of the lecture of December 2, 2022.
Recording and whiteboard of the lecture of November 29, 2022.
Recording and whiteboard of the lecture of November 25, 2022.
Recording and whiteboard of the lecture of November 22, 2022.
Recording and whiteboard of the lecture of November 18, 2022.
Recording and whiteboard of the lecture of November 15, 2022.
Recording and whiteboard of the lecture of November 11, 2022.
Recording and whiteboard of the lecture of November 8, 2022.
Recordings of year 2021
Recording and whiteboard of the seminar of December 21, 2021.
Recording and whiteboard of the lecture of December 17, 2021.
Recording and whiteboard of the lecture of December 14, 2021.
Recording and whiteboard of the lecture of December 10, 2021.
Recording and whiteboard of the lecture of December 7, 2021.
Recording and whiteboard of the lecture of December 3, 2021.
Recording and whiteboard of the lecture of November 30, 2021.
Recording and whiteboard of the lecture of November 26, 2021.
Recording and whiteboard of the lecture of November 23, 2021.
Recording and whiteboard of the lecture of November 19, 2021.
Recording and whiteboard of the lecture of November 16, 2021.
Recording and whiteboard of the lecture of November 12, 2021.
Recording and whiteboard of the lecture of November 9, 2021.
Recordings of year 2020
First, second part and whiteboard of the lecture of October 27, 2020.
First, second part and whiteboard of the lecture of October 20, 2020.
First, second part and whiteboard of the lecture of October 19, 2020.
First, second part and whiteboard of the lecture of October 13, 2020.
First, second part and whiteboard of the lecture of October 12, 2020.
First, second part and whiteboard of the lecture of October 6, 2020.
First, second part and whiteboard of the lecture of October 5, 2020.
First, second part and whiteboard of the lecture of September 29, 2020.
First, second part and whiteboard of the lecture of September 28, 2020.
First, second part and whiteboard of the lecture of September 22, 2020.
First, second, third part and whiteboard of the lecture of September 21, 2020.
Recordings of year 2019
First, second part and blackboard of the lecture of December 9, 2019.
First, second part and blackboard of the lecture of December 5, 2019.
First, second, third part and blackboard of the lecture of December 2, 2019.
First, second part and blackboard of the lecture of November 28, 2019.
First, second, third part and blackboard of the lecture of November 25, 2019.
First, second part and blackboard of the lecture of November 21, 2019.
First, second, third part and blackboard of the lecture of November 18, 2019.
First, second part and blackboard of the lecture of November 14, 2019.
First, second part and blackboard of the lecture of November 11, 2019.
First, second part and blackboard of the lecture of November 7, 2019.
First, second, third part and blackboard of the lecture of November 4, 2019. | {"url":"http://www.dm.unibo.it/~ferri/hm/progmame.htm","timestamp":"2024-11-05T12:46:45Z","content_type":"text/html","content_length":"24942","record_id":"<urn:uuid:2a00fd09-f42f-4a24-bdec-63efa3817600>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00394.warc.gz"} |
Kusmin-Landau Inequality
Let $I$ be the half-open interval $\hointl a b$.
Let $f: I \to R$ be continuously differentiable.
Let $f'$ be monotonic.
Let $\norm {f'} \ge \lambda$ on $I$ for some $\lambda \in \R_{>0}$, where $\norm {\, \cdot \,}$ denotes the distance to nearest integer.
$\ds \sum_{n \mathop \in I} e^{2 \pi i \map f n} = \map \OO {\frac 1 \lambda}$
where the big-$\OO$ estimate does not depend on $f$.
This article, or a section of it, needs explaining.
In particular: What does the sum over an interval $I$ mean?
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.
This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.
Source of Name
This entry was named for Rodion Osievich Kuzmin and Edmund Georg Hermann Landau. | {"url":"https://proofwiki.org/wiki/Kusmin-Landau_Inequality","timestamp":"2024-11-08T15:37:00Z","content_type":"text/html","content_length":"44719","record_id":"<urn:uuid:24406465-fa16-4b62-9c9b-02a6d9bcfc83>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00433.warc.gz"} |
Wavelet Archives - Machine Learning Applications
If we see in the real world, we will always face the signals which are not changing their stats. Means the change in the data signals are quite slow. But if we compare the 1D- data to the 2D Image
data then we can see the 2D images have more drastic change in the magnitude of the pixels due to edges, change in the contrast and the two different things in the same image.
Fourier Transform isn’t Able to Represent the Abrupt Changes Efficiently
So 1D data have slow oscillation but the images have more abrupt changes. These abrupt changing parts are always the interesting for that data as well as the images. They always show more relevant
information for the images and the data.
Now, we have great tool for the analysis of the signals and that is the Fourier transform. But, it doesn’t able to represent the abrupt changes efficiently. That’s the demerit of the Fourier
transform. The reason for this is that the Fourier transform is made up from the summation of the weighted sin and cosine signals. So, for abrupt changes that transform is less efficient.
Wavelets and Wavelet Transform is Great Tool for Abrupt Data Analysis
For that problem we must find out different bases except the sin and cosine because these bases are not efficient for the abrupt representation. For the solution of these problems, another great tool
came and those are the Wavelets and Wavelet transform. A wavelet is the rapidly decaying, wave like oscillation and that is also for the finite duration not like the sin and cosine (They oscillates
There are number of wavelets and based on the application and on the nature of the data, we can select the wavelet for that application and the data. Here, I have shown some of the well-known types
of wavelets.
Figure 1. Well known types of wavelets (Image is from MathWorks)
Now we are going to plot the Morlet in the MATLAB and that is quite easy if you know the basics of the MATLAB.
The equation for the Morlet wavelet is,
Let us plot the Morlet function using MATLAB.
%% Morlet Wavelet functions
lb = -4;% lower bound
ub = 4;% uper bound
n = 1000; % number of points
x = linspace(lb,ub,n);
y = exp(((-1)*(x.^2))./2).*cos(5*x);
% title(['Morlet Wavelet']);
title('Morlet Wavelet $$\psi(t) = e^{\frac{-x^2}{2}} \cos(5x)$$','interpreter','latex')
If we plot this wave then we will get the result like below,
Figure 2. Plot of the Morlet in the MATLAB.
We can see that this Morlet can able to represent the drastic changes and we can scale it for more drastic changes like Figure 3(b).
Figure 3a: Less abrupt change and the signal is applied as it is.
More abrupt change
Figure 3b: More abrupt change than the figure 3a. and in this case the signal is applied after some scaling.
Figure 3c: More abrupt change than the figure 3b. and in this case the signal is applied after very high scaling to represent the very sharp abrupt change.
Now, we have understood what exactly the Wavelets are. These wavelets are the bases for the Wavelet Transform similar like Sine and Cosines are the bases for the Fourier Transform.
The Wavelet Transform
The wavelet transform is the mathematical tool that can able to decomposes a signal into a representation of the signal’s fine details and the trends as the function of time. We can use this
transform or this representation to characterize the abrupt changes or transient events, to denoise, to perform many more operations on that.
The main benefit of wavelet transform or methods over traditional Fourier transform or methods are the uses of localized basis functions called as the wavelets and it give more faster computation.
Wavelets as being localized basis functions are best for analyzing real physical situations in which a signal have discontinuities, abrupt changes and sharp spikes.
Two major transforms that are very useful to wavelet analysis are the
Continuous Wavelet Transform
Discrete Wavelet Transform
If we see this equation then we will get feel like, oh!! That is very similar to the Fourier transform. Yes, that is very similar to that but here major difference is that ψ(t) and that is the
wavelet not the sin and the cosine. Here as a ψ(t), we can take any wavelet that suit best for our applications. Now we will be going to discuss about the uses of the wavelet transform.
The following are applications of wavelet transforms:
Data and image compression
Transient detection
Pattern recognition
Texture analysis
Noise/trend reduction
Wavelet Denoising
In this article we will go through the one application of the wavelet transform and that is denoising of 1-D data.
1-D Data:
I have taken the electrical data through the MATLAB.
load leleccum;
I have taken only the some part of that signal for the process.
s = leleccum(1:3920);
Figure 4. Electrical Signal Lelecum from the MATLAB.
This signal have so much sharp and abrupt changes and we can see some additional noise as well from 2500 to 3500. Here we can use the wavelet transform to denoise this signal.
First, we will perform only the one step Wavelet Decomposition of a Signal. For one step we will get only the two components and one will be approximation and the second will be the detail of the
signal. Here I have used the Daubechies wavelet for the wavelet transform.
[cA1,cD1] = dwt(s,’db1′);
This generates the coefficients of the level 1 approximation (cA1) and detail (cD1). This both are coefficients now we can construct the level 1 approximation and the detail as well.
A1 = upcoef('a',cA1,'db1',1,ls);
D1 = upcoef('d',cD1,'db1',1,ls);
If we display it then it will look something like Figure 5. We can see the approximation which are more and less similar to the signal and the details shows the sharp fluctuations of the signal.
Now, we will perform the decomposition of the signal in 3 levels. This decomposition will be the similar to the Figure 6. We can decompose the signal in these levels for more levels of details. Here
we will get three level details cD1, cD2 and cD3 and one approximation cA3.
We can create this 3 level decomposition using the “wavedec” function from the MATLAB. This function used for the decomposition of the signal in to multi-level wavelet decomposition.
[C,L] = wavedec(s,3,’db1′);
Here also I have used the Daubechies wavelet. The coefficients of all the components of a third-level decomposition (that is, the third-level approximation and the first three levels of detail) are
returned concatenated into one vector, C. Vector L gives the lengths of each component.
Figure 5. Approximation A1 and detail D1 at the first step.
Figure 6. Approximation and the details of the signal till level 3 (Image is from the MATHWORK).
We can extract the level 3 approximation coefficients from C using the “appcoef” function from the MATLAB.
cA3 = appcoef(C,L,'db1',3);
We can extract the level 3 details coefficients from C and L using the “detcoef” function from the MATLAB.
cD3 = detcoef(C,L,3);
cD2 = detcoef(C,L,2);
cD1 = detcoef(C,L,1);
This way we have total three values cA3, cD1, cD2, and cD3. We can reconstruct the approximate and details signals from these coefficients using “wrcoef”.
% To reconstruct the level 3 approximation from C,
A3 = wrcoef('a',C,L,'db1',3);
% To reconstruct the details at levels 1, 2 and 3,
D1 = wrcoef('d',C,L,'db1',1);
D2 = wrcoef('d',C,L,'db1',2);
D3 = wrcoef('d',C,L,'db1',3);
If we display this images then it will look something like Figure 7.
Figure 7. Approximation and details at the different levels.
We can use the wavelets to remove noise from a signal but it will requires identifying which component or components have the noise and then recovering the signal without those components. In this
example, we have observed that as we increase the number of the steps, the successive approximations become much less and less noisy because more and more high-frequency information is filtered out
of the signal.
If we compare the level 3 approximation with the original signal then we can find that level 3 approximation is much more smother than the original signal.
Of course, after removing all the high-frequency information, we will have lost many abrupt information from the original signal. So for optimal de-noising will required a more subtle method and that
is called as thresholding. Thresholding involves removing the portion from the details which have higher activity than the certain limits.
What if we limited the strength of the details by restricting their maximum values? This would have the effect of cutting back the noise while leaving the details unaffected through most of their
durations. But there’s a better way. We could directly manipulate each vector, setting each element to some fraction of the vectors’ peak or average value. Then we could reconstruct new detail
signals D1, D2, and D3 from the thresholded coefficients.
To denoise the image,
[thr,sorh,keepapp] = ddencmp('den','wv',s);
clean = wdencmp('gbl',C,L,'db1',3,thr,sorh,keepapp);
“ddencmp” function gives the default values of the threshold, SORH and KEEPAPP which allows you to keep approximation coefficients. Clean is the denoised signal.
Figure 8. Shows both the original as well as the clean signal.
Figure 8. Original signal with the De-noised signal.
Wavelet are the great tools for the analysis of the signals and those signals have ability to representation the signal in the great detail. Here we have experimented with the denoising of the
electrical signal, we have seen that using only low pass filter may affect the abrupt information of signals. But using the proper process of the wavelet transform we can have great denoised signal.
Wavelets can do much more than the denoising. Popular “.JPEG” encoding format for the images uses the discrete cosine transform for the compression of the images. There is other algorithm JPEG2000
which have great accuracy of the image with great compression. And JPEG2000 algorithm uses the wavelet transform.
Thus wavelets are very useful, so have great time with number of wavelets and may this article helps you to for the understanding of the wavelets.
For whole code in MATLAB and for more exciting projects please visit github.com/MachineLearning-Nerd/Wavelet-Analysis GITHUB repository. | {"url":"http://intelligentonlinetools.com/blog/category/wavelet/","timestamp":"2024-11-13T01:58:39Z","content_type":"text/html","content_length":"177808","record_id":"<urn:uuid:97da6eca-fdbc-40ed-8142-4d1aac0e4d92>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00887.warc.gz"} |
How to Perform Mann-Whitney U Test in Python with Scipy and Pingouin
In this data analysis tutorial, you will learn how to carry out a Mann-Whitney U test in Python with the packages SciPy and Pingouin. This test is also known as Mann–Whitney–Wilcoxon (MWW), Wilcoxon
rank-sum test, or Wilcoxon–Mann–Whitney test and is a non-parametric hypothesis test.
Table of Contents
Outline of the Post
This tutorial will teach you when and how to use this non-parametric test. After that, we will see an example of a situation when the Mann-Whitney U test can be used. The example is followed by how
to install the needed package (i.e., SciPy) as well as a package that makes importing data easy and that we can quickly visualize the data to support the interpretation of the results. In the
following section, you will learn the two steps to carry out the Mann-Whitney-Wilcoxon test in Python. Note we will also look at another package, Pingouin, that enables us to carry out statistical
tests with Python. Finally, we will learn how to interpret the results and visualize data to support our interpretation.
When to use the Mann-Whitney U test
This test is a rank-based test that can be used to compare values for two groups. If we get a significant result it suggests that the values for the two groups are different. As previously
mentioned, the Mann-Whitney U test is equivalent to a two-sample Wilcoxon rank-sum test.
Furthermore, we don’t have to assume that our data follows the normal distribution and can decide whether the population distributions are identical. Now, the Mann–Whitney test does not address
hypotheses about the medians of the groups. Rather, the test addresses whether an observation in one group is likely greater than an observation in the other group. In other words, it concerns
whether one sample has stochastic dominance compared with the other.
The test assumes that the observations are independent. That is, it is not appropriate for paired observations or repeated measures data.
Appropriate data
• One-way data with two groups: two-sample data, that is,
• Your dependent variable is of one of the three following: 1) ordinal, 2) interval, or 3) ratio,
• The independent variable is a factor with two levels (again, only two groups, see the first point),
• Observations between groups are independent. That is, not paired or repeated measures data
• To be a test of medians, the distributions of values for both the groups have to be of similar shape and spread. Under other conditions, the Mann-Whitney U test is by and large a test of
stochastic equality.
As with the two samples t-test there are normally two hypothesis:
• Null hypothesis (H[0]): The two groups are sampled from populations with identical distributions. Typically, the sampled populations exhibit stochastic equality.
• Alternative hypothesis (H[a]: The two groups are sampled from populations with different distributions (see the previous section). This usually means that one sampled population (group) displays
stochastic dominance.
If the results are significant, they can be reported as “The values for men were significantly different from those for women.”, if you are examining differences in values between men and women.
When do you use Mann-Whitney U Test?
You can use the Mann-Whitney U test when your outcome/dependent variable is ordinal or continuous but not normally distributed. Furthermore, this non-parametric test is used when you want to compare
differences between two independent groups (e.g., as an alternative to the two-sample t-test).
To conclude, you should use this test instead of e.g., two-sample t-test using Python if the above information is true for your data.
In this section, before moving on to how to carry out the test, we will have a quick look at an example of when you should use the Mann-Whitney U test.
If you, for example, run an intervention study designed to examine the effectiveness of a new psychological treatment to reduce symptoms of depression in adults. Let’s say that you have a total of n=
14 participants. Furthermore, these participants are randomized to receive either the treatment or no treatment. In your study, the participants are asked to record the number of depressive episodes
over one week following receipt of the assigned treatment. Here are some example data:
Example data
In this example, the question you might want to answer is: is there a difference in the number of depressive episodes over a 1 week period in participants receiving the new treatment as in comparison
to those receiving no treatment? By inspecting your data, it appears that participants receiving no treatment have more depressive episodes. The crucial question is, however, is this statistically
In this example, the outcome variable is the number of episodes (count), and, naturally, in this sample, the data do not follow a normal distribution. Note, Pandas was used to create the above
To follow this tutorial, you must have Pandas and SciPy installed. Now, you can get these packages using your favorite Python package manager. For example, installing Python packages with pip can be
done as follows:
pip install scipy pandas pingouinCode language: Bash (bash)
Note both Pandas and Pingouin are optional. However, using these packages has, as you will see later, their advantages. Hint, Pandas make data importing easy. If you ever need it, you can also use
pip to install a specific version of a package.
2 Steps to Perform the Mann-Whitney U test in Python
In this section, we will go through the steps to carry out the Mann-Whitney U test using Pandas and SciPy. In the first step, we will get our data. After storing the data in a dataframe, we will
carry out the non-parametric test.
Step1: Get your Data
Here’s one way to import data to Python with Pandas:
import pandas as pd
# Getting our data in to a dictionary
data = {'Notrt':[7, 5, 6, 4, 12, 9, 8],
'Trt':[3, 6, 4, 2, 1, 5, 1]}
# Dictionary to Dataframe
df = pd.DataFrame(data)Code language: Python (python)
In the code chunk above, we created a Pandas dataframe from a dictionary. Of course, most of the time, we will store our data in formats such as CSV or Excel.
See the following posts about how to import data in Python with Pandas:
Here’s also worth noting that if your data is stored in long format, you will have to subset your data such that you can get the data from each group into two different variables.
Step 2: Use the mannwhitneyu method from SciPy:
Here’s how to perform the Mann-Whitney U test in Python with SciPy:
from scipy.stats import mannwhitneyu
# Carrying out the Wilcoxon–Mann–Whitney test
results = mannwhitneyu(df['Notrt'], df['Trt'])
resultsCode language: Python (python)
Notice that we selected the columns, for each group, as x and y parameters to the mannwhitneyu method. If your data, as previously mentioned, is stored in long format (e.g., see image further down
below) you can use Pandas query() method to subset the data.
results from the wilcoxon rank sum test
Here’s how to perform the test using df.query(), if your data is stored in a similar way as in the image above:
import pandas as pd
idrt = [i for i in range(1,8)]
idrt += idrt
data = {'Count':[7, 5, 6, 4, 12, 9, 8,
3, 6, 4, 2, 1, 5, 1],
'Condition':['No Treatment']*7 + ['Treatment']*7, 'IDtrt':idrt}
# Dictionary to Dataframe
df = pd.DataFrame(data)
# Subsetting (i.e., creating new variables):
x = df.query('Condition == "No Treatment"')['Count']
y = df.query('Condition == "Treatment"')['Count']
# Mann-Whitney U test:
mannwhitneyu(x, y)Code language: Python (python)
Now, there are some things to be explained here. First, the mannwhitneyu method will by default carry out a one-sided test. On the other hand, if we would use the parameter alternative and set it to
“two-sided” we would get different results. Make sure you check out the documentation before using the method. In the next section, we will look at another, previously mentioned, Python package that
can also be used to do the Mann-Whitney U test.
Mann-Whitney U Test with the Python Package Pingouin
As previously mentioned, we can also install the Python package Pingouin to carry out the Mann-Whitney U test. Here’s how to perform this test with the mwu() method:
from pingouin import mwu
results2 = mwu(df['Notrt'], df['Trt'],
tail='one-sided')Code language: Python (python)
Now, the advantage with using the mwu method is that we will get some additional information (e.g., common language effect size; CLES). Here’s the output:
Interpreting the Results of the Mann-Whitney U test
In this section, we will start by interpreting the test results. Now, this is pretty straightforward.
In our example, we can reject H[0] because 3 < 7. Furthermore, we have statistically significant evidence at α =0.05 to show that the treatment groups differ in the number of depressive episodes.
Naturally, in a real application, we would have set both the H[0] and H[a] before conducting the hypothesis test, as we did here.
Visualizing the Data with Boxplots
To aid the interpretation of our results, we can create box plots with Pandas:
axarr = df.boxplot(column='Count', by='Condition',
figsize=(8, 6), grid=False)
axarr.set_ylabel('Number of Depressive Episodes')Code language: Python (python)
In the box plot, we can see that the median is greater for the group that did not get any treatment than the group that got treatment. Furthermore, if there were any outliers in our data they would
show up as dots in the box plot. If you are interested in more data visualization techniques, look at the post “9 Data Visualization Techniques You Should Learn in Python”.
Visualizing the results of Mann-Whitney U test
In this post, you have learned how to perform the Mann-Whitney U test using the Python packages SciPy, Pandas, and Pingouin. Moreover, you have learned when to carry out this non-parametric test by
learning about e.g., when it is appropriate and by an example. After this, you learned how to conduct the test using data from the example. Finally, you have learned how to interpret the results and
visualize the data. Note that you preferably should have a larger sample size than in the example of the current post. Of course, you should also decide whether to carry out a one-sided or two-sided
test based on theory. In the example of this post, we can assume that going without treatment would mean more depressive episodes. However, in other examples, this may not be true.
I hope you have learned something, and if you have a comment, a suggestion, or anything, you can leave a comment below. Finally, I would very much appreciate it if you shared this post across your
social media accounts if you found it useful!
In this final section, you will find some references and resources that may prove useful. Note, there are both links to blog posts and peer-reviewed articles. Sadly, some of the content here is
behind paywalls.
Mann, H. B.; Whitney, D. R. On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other. Ann. Math. Statist. 18 (1947), no. 1, 50–60. doi:10.1214/aoms/1177730491. https:/
Vargha, A., & Delaney, H. D. (2000). A Critique and Improvement of the CL Common Language Effect Size Statistics of McGraw and Wong. Journal of Educational and Behavioral Statistics, 25(2), 101–132.
Here are other Python tutorials that are helpful:
2 thoughts on “How to Perform Mann-Whitney U Test in Python with Scipy and Pingouin”
1. mamaligadoc
With respect !!!
1. Erik Marsja
Thanks for your comment.
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Towards physics-inspired data-driven weather forecasting: integrating data assimilation with a deep spatial-transformer-based U-NET in a case study with ERA5
Articles | Volume 15, issue 5
© Author(s) 2022. This work is distributed under the Creative Commons Attribution 4.0 License.
Towards physics-inspired data-driven weather forecasting: integrating data assimilation with a deep spatial-transformer-based U-NET in a case study with ERA5
There is growing interest in data-driven weather prediction (DDWP), e.g., using convolutional neural networks such as U-NET that are trained on data from models or reanalysis. Here, we propose three
components, inspired by physics, to integrate with commonly used DDWP models in order to improve their forecast accuracy. These components are (1) a deep spatial transformer added to the latent space
of U-NET to capture rotation and scaling transformation in the latent space for spatiotemporal data, (2) a data-assimilation (DA) algorithm to ingest noisy observations and improve the initial
conditions for next forecasts, and (3) a multi-time-step algorithm, which combines forecasts from DDWP models with different time steps through DA, improving the accuracy of forecasts at short
intervals. To show the benefit and feasibility of each component, we use geopotential height at 500hPa (Z500) from ERA5 reanalysis and examine the short-term forecast accuracy of specific setups of
the DDWP framework. Results show that the spatial-transformer-based U-NET (U-STN) clearly outperforms the U-NET, e.g., improving the forecast skill by 45%. Using a sigma-point ensemble Kalman
(SPEnKF) algorithm for DA and U-STN as the forward model, we show that stable, accurate DA cycles are achieved even with high observation noise. This DDWP+DA framework substantially benefits from
large (O(1000)) ensembles that are inexpensively generated with the data-driven forward model in each DA cycle. The multi-time-step DDWP+DA framework also shows promise; for example, it reduces the
average error by factors of 2–3. These results show the benefits and feasibility of these three components, which are flexible and can be used in a variety of DDWP setups. Furthermore, while here we
focus on weather forecasting, the three components can be readily adopted for other parts of the Earth system, such as ocean and land, for which there is a rapid growth of data and need for forecast
and assimilation.
Received: 11 Mar 2021 – Discussion started: 12 Apr 2021 – Revised: 16 Feb 2022 – Accepted: 19 Feb 2022 – Published: 16 Mar 2022
Motivated by improving weather and climate prediction, using machine learning (ML) for data-driven spatiotemporal forecasting of chaotic dynamical systems and turbulent flows has received substantial
attention in recent years (e.g., Pathak et al., 2018; Vlachas et al., 2018; Dueben and Bauer, 2018; Scher and Messori, 2018, 2019; Chattopadhyay et al., 2020b, c; Nadiga, 2020; Maulik et al., 2021).
These data-driven weather prediction (DDWP) models leverage ML methods such as convolutional neural networks (CNNs) and/or recurrent neural networks (RNNs) that are trained on state variables
representing the history of the spatiotemporal variability and learn to predict the future states (we have briefly described some of the technical ML terms in Table 1). In fact, a few studies have
already shown promising results with DDWP models that are trained on variables representing the large-scale circulation obtained from numerical models or reanalysis products (Scher, 2018;
Chattopadhyay et al., 2020a; Weyn et al., 2019, 2020; Rasp et al., 2020; Arcomano et al., 2020; Rasp and Thuerey, 2021). Chattopadhyay et al. (2020d) showed that DDWP models trained on general
circulation model (GCM) outputs can be used to predict extreme temperature events. Excellent reviews and opinion pieces on the state of the art of DDWP can be found in Chantry et al. (2021),
Watson-Parris (2021), and Irrgang et al. (2021). Other applications of DDWP may include post-processing of ensembles (Grönquist et al., 2021) and sub-seasonal to seasonal prediction (Scher and
Messori, 2021; Weyn et al., 2021).
(Lütkepohl, 2013)(Goodfellow et al., 2016)(Goodfellow et al., 2016)(Evensen, 1994)(Wang et al., 2020; Bronstein et al., 2021)(Goodfellow et al., 2016)(Tang et al., 2014)(Jaderberg et al., 2015)(Wan
et al., 2001)
The increasing interest (Schultz et al., 2021; Balaji, 2021) in these DDWP models stems from the hope that they improve weather forecasting because of one or both of the following reasons: (1)
trained on reanalysis data and/or data from high-resolution NWP models, these DDWP models may not suffer from some of the biases (or generally, model error) of physics-based, operational numerical
weather prediction (NWP) models, and (2) the low computational cost of these DDWP models allows for generating large ensembles for probabilistic forecasting (Weyn et al., 2020, 2021). Regarding
reason (1), while DDWP models trained on reanalysis data have skills for short-term predictions, so far they have not been able to outperform operational NWP models (Weyn et al., 2020; Arcomano
et al., 2020; Schultz et al., 2021). This might be, at least partly, due to the short training sets provided by around 40 years of high-quality reanalysis data (Rasp and Thuerey, 2021). There are a
number of ways to tackle this problem; for example, transfer learning could be used to blend data from low- and high-fidelity data or models (e.g., Ham et al., 2019; Chattopadhyay et al., 2020e; Rasp
and Thuerey, 2021), and/or physical constraints could be incorporated into the often physics-agnostic ML models, which has been shown in applications of high-dimensional fluid dynamics (Raissi et al.
, 2020) as well as toy examples of atmospheric or oceanic flows (Bihlo and Popovych, 2021). The first contribution of this paper is to provide a framework for the latter, by integrating the
convolutional architectures with deep spatial transformers that capture rotation, scaling, and translation within the latent space that encodes the data obtained from the system. The second
contribution of this paper is to equip these DDWP models with data assimilation (DA), which provides improved initial conditions for weather forecasting and is one of the key reasons behind the
success of NWP models. Below, we further discuss the need for integrating DA with DDWP models which can capture rotation and scaling transformations in the flow and briefly describe what has been
already done in these areas in previous studies.
Many of the DDWP models built so far are physics agnostic and learn the spatiotemporal evolution only from the training data, resulting sometimes in physically inconsistent predictions and an
inability to capture key invariants and symmetries of the underlying dynamical system, particularly when the training set is small (Reichstein et al., 2019; Chattopadhyay et al., 2020d). There are
various approaches to incorporating some physical properties into the neural networks; for example, Kashinath et al. (2021) have recently reviewed 10 approaches (with examples) for physics-informed
ML in the context of weather and climate modeling. One popular approach, in general, is to enforce key conservation laws, symmetries, or some (or even all) of the governing equations through
custom-designed loss functions (e.g., Raissi et al., 2019; Beucler et al., 2019; Daw et al., 2020; Mohan et al., 2020; Thiagarajan et al., 2020; Beucler et al., 2021).
Another approach – which has received less attention particularly in weather and climate modeling – is to enforce the appropriate symmetries, which are connected to conserved quantities through
Noether's theorem (Hanc et al., 2004), inside the neural architecture. For instance, conventional CNN architectures enforce translational and rotational symmetries, which may not necessarily exist in
the large-scale circulation; see Chattopadhyay et al. (2020d) for an example based on atmospheric blocking events and rotational symmetry. Indeed, recent research in the ML community has shown that
preserving a more general property called “equivariance” can improve the performance of CNNs (Maron et al., 2018, 2019; Cohen et al., 2019). Equivariance-preserving neural network architectures learn
the existence of (or lack thereof) symmetries in the data rather than enforcing them a priori and better track the relative spatial relationship of features (Cohen et al., 2019). In fact, in their
work on forecasting midlatitude extreme-causing weather patterns, Chattopadhyay et al. (2020d) have shown that capsule neural networks, which are equivariance-preserving (Sabour et al., 2017),
outperform conventional CNNs in terms of out-of-sample accuracy while requiring a smaller training set. Similarly, Wang et al. (2020) have shown the advantages of equivariance-preserving CNN
architectures in data-driven modeling of Rayleigh–Bénard and ocean turbulence. More recently, using two-layer quasi-geostrophic turbulence as the test case, Chattopadhyay et al. (2020c) have shown
that capturing rotation, scaling, and translational features in the flow in the latent space of a CNN architecture through a deep-spatial-transformer architecture (Jaderberg et al., 2015) improves
the accuracy and stability of the DDWP models without increasing the network's complexity or computational cost (which are drawbacks of capsule neural networks). Building on these studies, here our
first goal is to develop a physics-inspired, autoregressive DDWP model that uses a deep spatial transformer in an encoder–decoder U-NET architecture (Ronneberger et al., 2015). Note that our approach
to use a deep spatial transformer is different from enforcing invariants in the loss function in the form of partial differential equations of the system (Raissi et al., 2019).
DA is an essential component of modern weather forecasting (e.g., Kalnay, 2003; Carrassi et al., 2018; Lguensat et al., 2019). DA corrects the atmospheric state forecasted using a forward model
(often a NWP model) by incorporating noisy and partial observations from the atmosphere (and other components of the Earth system), thus estimating a new corrected state of the atmosphere called
“analysis”, which serves as an improved initial condition for the forward model to forecast the future states. Most operational forecasting systems have their NWP model coupled to a DA algorithm that
corrects the trajectory of the atmospheric states, e.g., every 6h with observations from remote sensing and in situ measurements. State-of-the-art DA algorithms use variational and/or ensemble-based
approaches. The challenge with the former is computing the adjoint of the forward model, which involves high-dimensional, nonlinear partial differential equations (Penny et al., 2019). Ensemble-based
approaches, which are usually variants of ensemble Kalman filter (EnKF; Evensen, 1994), bypass the need for computing the adjoint but require generating a large ensemble of states that are each
evolved in time using the forward model, which makes this approach computationally expensive (Hunt et al., 2007; Houtekamer and Zhang, 2016; Kalnay, 2003).
In recent years, there has been a growing number of studies at the intersection of ML and DA (Geer, 2021). A few studies have aimed, using ML, to accelerate and improve DA frameworks, e.g., by taking
advantage of their natural connection (Abarbanel et al., 2018; Kovachki and Stuart, 2019; Grooms, 2021; Hatfield et al., 2021). A few other studies have focused on using DA to provide suitable
training data for ML from noisy or sparse observations (Brajard et al., 2020, 2021; Tang et al., 2020; Wikner et al., 2021). Others have integrated DA with a data-driven or hybrid forecast model for
relatively simple dynamical systems (Hamilton et al., 2016; Lguensat et al., 2017; Lynch, 2019; Pawar and San, 2020). However, to the best of our knowledge, no study has yet integrated DA with a DDWP
model. Here, our second goal is to present a DDWP+DA framework in which the DDWP is the forward model that efficiently provides a large, O(1000), ensemble of forecasts for a sigma-point ensemble
Kalman filter (SPEnKF) algorithm.
To provide proofs of concept for the DDWP model and the combined DDWP+DA framework, we use sub-daily 500hPa geopotential height (Z500) from the ECMWF Reanalysis 5 (ERA5) dataset (Hersbach et al.,
2020). The DDWP model is trained on hourly, 6, or 12h Z500 samples. The spatiotemporal evolution of Z500 is then forecasted from precise initial conditions using the DDWP model or from noisy initial
conditions using the DDWP+SPEnKF framework. Our main contributions in this paper are three-fold, namely,
• Introducing the spatial-transformer-based U-NET that can capture rotational and scaling features in the latent space for DDWP modeling and showing the advantages of this architecture over a
conventional encoder–decoder U-NET.
• Introducing the DDWP+DA framework, which leads to stable DA cycles without the need for any localization or inflation by taking advantage of the large forecast ensembles produced in a data-driven
fashion using the DDWP model.
• Introducing a novel multi-time-step method for improving the DDWP+DA framework. This framework utilizes virtual observations produced using more accurate DDWP models that have longer time steps.
This framework exploits the non-trivial dependence of the accuracy of autoregressive data-driven models on the time step size.
The remainder of the paper is structured as follows. The data are described in Sect. 2. The encoder–decoder U-NET architecture with the deep spatial transformer and the SPEnKF algorithm are
introduced in Sect. 3. Results are presented in Sect. 4, and the discussion and summary are in Sect. 5.
We use the ERA5 dataset from the WeatherBench repository (Rasp et al., 2020), where each global sample of Z500 at every hour is downsampled to a rectangular longitude–latitude (x,y) grid of 32×64. We
have chosen the variable Z500 following previous work (Weyn et al., 2019, 2020; Rasp et al., 2020) as an example, because it is representative of the large-scale circulation in the troposphere and
influences near-surface weather and extremes. This coarse-resolution Z500 dataset from the WeatherBench repository has been used in a number of recent studies to perform data-driven weather
forecasting (Rasp et al., 2020; Rasp and Thuerey, 2021). Here, we use Z500 data from 1979 to 2015 (≈315360 samples) for training, 2016–2017 (≈17520 samples) for validation, and 2018 (≈8760 samples)
for testing.
3.1The spatial-transformer-based DDWP model: U-NET with a deep spatial transformer (U-STN)
The DDWP models used in this paper are trained on Z500 data without access to any other atmospheric fields that might affect the atmosphere's spatiotemporal evolution. Once trained on past Z500
snapshots sampled at every Δt, the DDWP model takes Z500 at a particular time t (Z(t) hereafter) as the input and predicts Z(t+Δt), which is then used as the input to predict Z(t+2Δt), and this
autoregressive process continues as needed. We use Δt, i.e., 1, 6, or 12h. The baseline DDWP model used here is a U-NET similar to the one used in Weyn et al. (2020). For the DDWP introduced here,
the encoded latent space of the U-NET is coupled with a deep spatial transformer (U-STN hereafter) to capture rotational and scaling features between the latent space and the decoded output. The
spatial-transformer-based latent space tracks translation, rotation, and stretching of the synoptic- and larger-scale patterns, and it is expected to improve the forecast of the spatiotemporal
evolution of the midlatitude Rossby waves and their nonlinear breaking. In this section, we briefly discuss the U-STN architecture, which is schematically shown in Fig. 1. Note that from now on “x”
in U-STNx (and U-NETx) indicates the Δt (in hours) that is used; for example, U-STN6 uses Δt=6h.
3.1.1Localization network or encoding block of U-STN
The network takes in an input snapshot of Z500, Z(t)^32×64, as initial condition and projects it onto a low-dimensional encoding space via a U-NET convolutional encoding block. This encoding block
performs two successive sets of two convolution operations (without changing the spatial dimensions) followed by a max-pooling operation. It is then followed by two convolutions without max pooling
and four dense layers. More details on the exact set of operations inside the architecture are reported in Table 2. The convolutions inside the encoder block account for Earth's longitudinal
periodicity by performing circular convolutions (Schubert et al., 2019) on each feature map inside the encoder block. The encoded feature map, which is the output of the encoding block and consists
of the reduced Z and coordinate system, ${\stackrel{\mathrm{̃}}{Z}}^{\mathrm{8}×\mathrm{16}}$ and $\left({x}_{i}^{\mathrm{o}},{y}_{i}^{\mathrm{o}}\right)$ where $i=\mathrm{1},\mathrm{2}\mathrm{\dots }
\mathrm{8}×\mathrm{16}$, is sent to the spatial transformer module described below.
3.1.2Spatial transformer module
The spatial transformer (Jaderberg et al., 2015) applies an affine transformation T(θ) to the reduced coordinate system $\left({x}_{i}^{\mathrm{o}},{y}_{i}^{\mathrm{o}}\right)$ to obtain a new
transformed coordinate system $\left({x}_{i}^{\mathrm{s}},{y}_{i}^{\mathrm{s}}\right)$:
$\begin{array}{}\text{(1)}& \left[\begin{array}{c}{x}_{i}^{\mathrm{s}}\\ {y}_{i}^{\mathrm{s}}\end{array}\right]=T\left(\mathit{\theta }\right)\left[\begin{array}{c}{x}_{i}^{\mathrm{o}}\\ {y}_{i}^{\
mathrm{o}}\\ \mathrm{1}\end{array}\right],\end{array}$
$\begin{array}{}\text{(2)}& T\left(\mathit{\theta }\right)=\left[\begin{array}{ccc}{\mathit{\theta }}_{\mathrm{11}}& {\mathit{\theta }}_{\mathrm{12}}& {\mathit{\theta }}_{\mathrm{13}}\\ {\mathit{\
theta }}_{\mathrm{21}}& {\mathit{\theta }}_{\mathrm{22}}& {\mathit{\theta }}_{\mathrm{23}}\end{array}\right].\end{array}$
The parameters θ are predicted for each sample. A differentiable sampling kernel (a bilinear interpolation kernel in this case) is then used to transform ${\stackrel{\mathrm{̃}}{Z}}^{\mathrm{8}×\
mathrm{16}}$, which is on the old coordinate system $\left({x}_{i}^{\mathrm{o}},{y}_{i}^{\mathrm{o}}\right)$, into ${\overline{Z}}^{\mathrm{8}×\mathrm{16}}$, which is on the new coordinate system $\
left({x}_{i}^{\mathrm{s}},{y}_{i}^{\mathrm{s}}\right)$. Note that in this architecture, the spatial transformer is applied to the latent space and its objective is to ensure that no a priori symmetry
structure is assumed in the latent space. The parameters in T(θ) learn the transformation (translation, rotation, and scaling) between the input to the latent space and the decoded output. It must be
noted here that this does not ensure that the entire network is equivariant by construction.
We highlight that in this paper we are focusing on capturing effects of translation, rotation, and scaling of the input field, because those are the ones that we expect to matter the most for the
synoptic patterns on a 2D plane. Furthermore, here we focus on an architecture with a transformer that acts only on the latent space. More complex architectures, with transformations like Eq. (1)
after every convolution layer, can be used too albeit with a significant increase in computational cost (de Haan et al., 2020; Wang et al., 2020). Our preliminary exploration shows that, for this
work, the one spatial transformer module applied on the latent space of the U-NET yields sufficiently superior performance (over the baseline, U-NET), but further exhaustive explorations should be
conducted in future studies to find the best-performing architecture for each application. Moreover, recent work in neural architecture search for geophysical turbulence shows that, with enough
computing power, one can perform exhaustive searches over optimal architectures, a direction that should be pursued in future work (Maulik et al., 2020).
Finally we point out that without the transformer module, $\overline{Z}=\stackrel{\mathrm{̃}}{Z}$, and the network becomes a standard U-NET.
3.1.3Decoding block
The decoding block is a series of deconvolution layers (convolution with zero-padded up-sampling) concatenated with the corresponding convolution outputs from the encoder part of the U-NET. The
decoding blocks bring the latent space ${\overline{Z}}^{\mathrm{8}×\mathrm{16}}$ back into the original dimension and coordinate system at time t+Δt, thus outputting $Z\left(t+\mathrm{\Delta }t{\
right)}^{\mathrm{32}×\mathrm{64}}$. The concatenation of the encoder and decoder convolution outputs allows the architecture to learn the features in the small-scale dynamics of Z500 better (Weyn
et al., 2020).
The loss function L to be minimized is
$\begin{array}{ll}L\left(\mathit{\lambda }\right)=& \phantom{\rule{0.25em}{0ex}}\frac{\mathrm{1}}{\left(N+\mathrm{1}\right)}\\ \text{(3)}& & ×\sum _{t=\mathrm{0}}^{t=N\mathrm{\Delta }t}||\left(Z\left
(t+\mathrm{\Delta }t\right)-\text{U-STNx}\left(Z\left(t\right),\mathit{\lambda }\right)\right)|{|}_{\mathrm{2}}^{\mathrm{2}},\end{array}$
where N is the number of training samples, t=0 is the start time of the training set, and λ represents the parameters of the network that are to be trained (in this case, the weights, biases, and θ
of U-STNx). In both encoding and decoding blocks, the rectified linear unit (ReLU) activation functions are used. The number of convolutional kernels (32 in each layer), size of each kernel (5×5),
Gaussian initialization, and the learning rate ($\mathit{\alpha }=\mathrm{3}×{\mathrm{10}}^{-\mathrm{4}}$) have been chosen after extensive trial and error. All codes for these networks (as well as
DA) have been made publicly available on GitHub and Zenodo (see the “Code and data availability” statement). A comprehensive list of information about each of the layers in both the U-STNx and U-NETx
architectures is presented in Table 2 along with the optimal set of hyperparameters that have been obtained through extensive trial and error.
Note that the use of U-NET is inspired from the work by Weyn et al. (2020); however, the architecture used in this study is different from that by Weyn et al. (2020). The main differences are in the
number of convolution layers and filters used in the U-NET along with the spatial transformer module. Apart from that, in Weyn et al. (2020) the mechanism by which autoregressive prediction is done
is different from this paper. Two time steps (6 and 12h) are predicted directly as the output by Weyn et al. (2020) using the U-NET. Moreover, the data for training and testing in Weyn et al. (2020)
are on the gnomonic cubed sphere.
3.2Data assimilation algorithm and coupling with DDWP
For DA, we employ the SPEnKF algorithm, which unlike the EnKF algorithm, does not use random perturbations to generate an ensemble but rather uses an unscented transformation (Wan et al., 2001) to
deterministically find an optimal set of points called sigma points (Ambadan and Tang, 2009). The SPEnKF algorithm has been shown to outperform EnKF on particular test cases for both chaotic
dynamical systems and ocean dynamics (Tang et al., 2014), although whether it is always superior to EnKF is a matter of active research (Hamill et al., 2009) and beyond the scope of this paper. Our
DDWP+DA framework can use any ensemble-based algorithm.
In the DDWP+DA framework, shown schematically in Fig. 2, the forward model is a DDWP, which is chosen to be U-STN1 and denoted as Ψ below. We use σ[obs] for the standard deviation of the observation
noise, which in this paper is either σ[obs]=0.5σ[Z] or σ[obs]=σ[Z], where σ[Z] is the standard deviation of Z500 over all grid points and over all years between 1979–2015. Here, we assume that the
noisy observations are assimilated every 24h (again, the framework can be used with any DA frequency, such as 6h, which is used commonly in operational forecasting).
We start with a noisy initial condition Z(t), and we use U-STN1 to autoregressively (with Δt=1h) predict the next time steps, i.e., Z(t+Δt), Z(t+2Δt), Z(t+3Δt), up to Z(t+23Δt). For a D-dimensional
system (i.e., Z∈ℝ^D), the optimal number of ensemble members for SPEnKF is 2D (Ambadan and Tang, 2009). Because here $D=\mathrm{32}×\mathrm{64}$, then 4096 ensemble members are needed. While this is
a very large ensemble size if the forward model is a NWP (operationally, ∼50–100 members are used; Leutbecher, 2019), the DDWP can inexpensively generate O(1000) ensemble members, a major advantage
of DDWP as a forward model that we will discuss later in Sect. 5.
To do SPEnKF, an ensemble of states at the 23rd hour of each DA cycle (24h is one DA cycle) is generated using a symmetric set of sigma points (Julier and Uhlmann, 2004) as
$\begin{array}{ll}{Z}_{\text{ens}}^{i}\left(t+\mathrm{23}\mathrm{\Delta }t\right)& =Z\left(t+\mathrm{23}\mathrm{\Delta }t\right)-{\mathbit{A}}_{i},\\ \text{(4)}& {Z}_{\text{ens}}^{j}\left(t+\mathrm
{23}\mathrm{\Delta }t\right)& =Z\left(t+\mathrm{23}\mathrm{\Delta }t\right)+{\mathbit{A}}_{j},\end{array}$
where $i,j\in \left[\mathrm{1},\mathrm{2},\mathrm{\cdots },D=\mathrm{32}×\mathrm{64}\right]$ are indices of the 2D ensemble members. Vectors A[i] and A[j] are columns of matrix $\mathbf{A}=\mathbf{U}
\sqrt{\mathbf{S}}{\mathbf{U}}^{\mathbf{T}}$, where U and S are obtained from the singular value decomposition of the analysis covariance matrix P[a], i.e., P[a]=USV^T. The D×D matrix P[a] is either
available from the previous DA cycle (see Eq. 10 below) or is initialized as an identity matrix at the beginning of DA. Note that here we generate the ensemble at one Δt before the next DA; however,
the ensembles can be generated at any time within the DA cycle and carried forward, although that would increase the computational cost of the framework. We have explored generating the ensembles at
t+0Δt (i.e., the beginning) but did not find any improvement over Eq. (4). It must however be noted that by not propagating the ensembles for 24h, the spread of the ensembles underestimates the
background error.
Once the ensembles are generated via Eq. (4), every ensemble member is fed into Ψ to predict an ensemble of forecasted states at t+24Δt:
$\begin{array}{}\text{(5)}& {Z}_{\text{ens}}^{k}\left(t+\mathrm{24}\mathrm{\Delta }t\right)=\mathbf{\Psi }\left({Z}_{\text{ens}}^{k}\left(t+\mathrm{23}\mathrm{\Delta }t\right)\right),\end{array}$
where $k\in \mathit{\left\{}-D,-D+\mathrm{1},\mathrm{\dots },D-\mathrm{1},D\mathit{\right\}}$. In general, the modeled observation is $\mathbf{H}\left(〈{Z}_{\text{ens}}^{k}\left(t+\mathrm{24}\mathrm
{\Delta }t\right)〉,\mathit{ϵ}\left(t\right)\right)$, where H is the observation operator, and ϵ(t) is the Gaussian random process with standard deviation σ[obs] that represents the uncertainty in
the observation. 〈.〉 denotes ensemble averaging. In this paper, we assume that H is the identity matrix while we acknowledge that, in general, it could be a nonlinear function. The SPEnKF algorithm
can account for such complexity, but here, to provide a proof of concept, we have assumed that we can observe the state, although with a certain level of uncertainty. With H=I, the background error
covariance matrix P[b] becomes
$\begin{array}{ll}{\mathbf{P}}_{\mathbf{b}}=\mathbf{E}\left[& \left({Z}_{\text{ens}}^{k}\left(t+\mathrm{24}\mathrm{\Delta }t\right)-〈{Z}_{\text{ens}}^{k}\left(t+\mathrm{24}\mathrm{\Delta }t\right)〉
\right)\\ \text{(6)}& & ×\left({Z}_{\text{ens}}^{k}\left(t+\mathrm{24}\mathrm{\Delta }t\right)-〈{Z}_{\text{ens}}^{k}\left(t+\mathrm{24}\mathrm{\Delta }t\right)〉{\right)}^{\mathbf{T}}\right],\end
where [.]^T denotes the transpose operator, and E[.] denotes the expectation operator. The innovation covariance matrix is defined as
$\begin{array}{}\text{(7)}& \mathbf{C}={\mathbf{P}}_{\mathbf{b}}+\mathbf{R},\end{array}$
where the observation noise matrix R is a constant diagonal matrix of the variance of observation noise, i.e., ${\mathit{\sigma }}_{\text{obs}}^{\mathrm{2}}$. The Kalman gain matrix is then given by
$\begin{array}{}\text{(8)}& \mathbf{K}={\mathbf{P}}_{\mathbf{b}}{\mathbf{C}}^{-\mathrm{1}},\end{array}$
and the estimated (analysis) state $\stackrel{\mathrm{^}}{Z}\left(t+\mathrm{24}\mathrm{\Delta }t\right)$ is calculated as
$\begin{array}{ll}\stackrel{\mathrm{^}}{Z}\left(t+\mathrm{24}\mathrm{\Delta }t\right)=& \phantom{\rule{0.25em}{0ex}}〈Z\left(t+\mathrm{24}\mathrm{\Delta }t\right)〉\\ \text{(9)}& & -\mathbf{K}\left
(〈{Z}_{\text{ens}}^{k}\left(t+\mathrm{24}\mathrm{\Delta }t\right)〉-{Z}^{\text{obs}}\left(t+\mathrm{24}\mathrm{\Delta }t\right)\right),\end{array}$
where Z^obs(t+24Δt) is the noisy observed Z500 at t+24Δt, i.e., ERA5 value at each grid point plus random noise drawn from $\mathcal{N}\left(\mathrm{0},{\mathit{\sigma }}_{\text{obs}}^{\mathrm{2}}\
right)$. While adding Gaussian random noise to the truth is an approximation, it is a quite common in the DA literature (Brajard et al., 2020, 2021; Pawar et al., 2020). The analysis error covariance
matrix is updated as
$\begin{array}{}\text{(10)}& {\mathbf{P}}_{\mathbf{a}}={\mathbf{P}}_{\mathbf{b}}-{\mathbf{KCK}}^{\mathbf{T}}.\end{array}$
The estimated state $\stackrel{\mathrm{^}}{Z}\left(t+\mathrm{24}\mathrm{\Delta }t\right)$ becomes the new initial condition to be used by U-STN1, and the updated P[a] is used to generate the
ensembles in Eq. (4) after another 23h for the next DA cycle.
Finally, we remark that often with low ensemble sizes, the background covariance matrix, P[b] (Eq. 6), suffers from spurious correlations which are corrected using localization and inflation
strategies (Hunt et al., 2007; Asch et al., 2016). However, due to the large ensemble size used here (with 4096 ensemble members that are affordable because of the computationally inexpensive DDWP
forward model), we do not need to perform any localization or inflation on P[b] to get stable DA cycles as shown in the next section.
4.1Performance of the spatial-transformer-based DDWP: noise-free initial conditions (no DA)
First, we compare the performance a U-STN and a conventional U-NET, where the only difference is in the use of the spatial transformer module in the former. Using U-STN12 and U-NET12 as
representatives of these architectures, Fig. 3 shows the anomaly correlation coefficients (ACCs) between the predictions from U-STN12 or U-NET12 and the truth (ERA5) for 30 noise-free, random initial
conditions. ACC is computed every 12h as the correlation coefficient between the predicted Z500 anomaly and the Z500 anomaly of ERA5, where anomalies are derived by removing the 1979–2015 time mean
of Z500 of the ERA5 dataset. U-STN12 clearly outperforms U-NET12, most notably after 36h, reaching ACC=0.6 after around 132h, a 45% (1.75d) improvement over U-NET12, which reaches ACC=0.6 after
around 90h.
To further see the source of this improvement, Fig. 4 shows the spatiotemporal evolution of Z500 patterns from an example of prediction using U-STN12 and U-NET12. Comparing with the truth (ERA5),
U-STN12 can better capture the evolution of the large-amplitude Rossby waves and the wave-breaking events compared to U-NET12; for example, see the patterns over Central Asia, Southern Pacific Ocean,
and Northern Atlantic Ocean on days 2–5. We cannot rigorously attribute the better capturing of wave-breaking events to an improved representation of physical features by the spatial transformer.
However, the overall improvement in performance of U-STN12 due to the spatial transformer (which is the only difference between U-STN12 and U-NET12) may lead to capturing some wave-breaking events in
the atmosphere as can be seen from exemplary evidence in Fig. 4. Furthermore, on days 4 and 5, the predictions from U-NET12 have substantially low Z500 values in the high latitudes of the Southern
Hemisphere, showing signs of unphysical drifts.
Overall, the results of Figs. 3 and 4 show the advantages of using the spatial-transformer-enabled U-STN in DDWP models. It is important to note that it is difficult to assert whether the
transformation with T(θ) in the latent space actually leads to physically meaningful transformations in the decoded output. However, we see that the performance of the network improves with the
addition of the spatial transformer module. Future studies need to focus on more interpretation of what the T(θ) matrix inside neural networks captures (Bronstein et al., 2021). Note that while here
we show results with Δt=12h, similar improvements are seen with Δt=1 and Δt=6h (see Sect. 4.3). Furthermore, to provide a proof of concept for the U-STN, in this paper we focus on Z500
(representing the large-scale circulation) as the only state variable to be learned and predicted. Even without access to any other information (e.g., about small scales), the DDWP model can provide
skillful forecasts for some time, consistent with earlier findings with the multi-scale Lorenz 96 system (Dueben and Bauer, 2018; Chattopadhyay et al., 2020b). More state variables can be easily
added to the framework, which is expected to extend the forecast skill, based on previous work with U-NET (Weyn et al., 2020). In this work, we have considered Z500 as an example for a proof of
concept. We have also performed experiments (not shown for brevity) by adding T850 as one of the variables to the input along with Z500 in U-NETx and U-STNx and found similarly good prediction
performance for the T850 variable.
A benchmark for different DDWP models has been shown in Rasp et al. (2020), with different ML algorithms such as CNN, linear regression, etc. In terms of RMSE for Z500 (Fig. 6, left panel, shows RMSE
of U-STNx and U-NETx in this paper with different Δt), U-STN12 outperforms the CNN model in WeatherBench (Rasp et al., 2020) by 33.2m at lead time of 3d and 26.7m at lead time of 5d. Similarly,
U-STN12 outperforms the linear regression in WeatherBench by 39.9m at lead time of 3d and by 29.3m at lead time of 5d. Note that in more recent work (Weyn et al., 2020; Rasp and Thuerey, 2021),
prediction horizons outperforming the WeatherBench models (Rasp et al., 2020) have also been shown.
4.2Performance of the DDWP+DA framework: noisy initial conditions and assimilated observations
To analyze the performance of the DDWP+DA framework, we use U-STN1 as the DDWP model and SPEnKF as the DA algorithm, as described in Sect. 3.2. In this U-STN1+SPEnKF setup, the initial conditions for
predictions are noisy observations and every 24h, noisy observations are assimilated to correct the forecast trajectory (as mentioned before, noisy observations are generated by adding random noise
from 𝒩(0,σ[obs]) to the Z500 of ERA5).
In Fig. 5, for 30 random initial conditions and two noise levels (σ[obs]=0.5σ[Z] or 1σ[Z]), we report the spatially averaged root-mean-square error (RMSE) and the correlation coefficient (R) of the
forecasted full Z500 fields as compared to the truth, i.e., the (noise-free) Z500 fields of ERA5. For both noise levels, we see that within each DA cycle, the forecast accuracy decreases between 0
and 23h until DA with SPEnKF occurs at the 24th hour, wherein information from the noisy observation is assimilated to improve the estimate of the forecast at the 24th hour. This estimate acts as
the new improved initial condition to be used by U-STN1 to forecast future time steps. In either case, the RMSE and R remain below 30m (80m) and above 0.7 (0.3) with σ[obs]=0.5σ[Z] (σ[obs]=1σ[Z])
for the first 10d. The main point here is not the accuracy of the forecast (which as mentioned before and could be further extended, e.g., by adding more state variables), but the stability of the
U-STN1+SPEnKF framework (without localization or inflation), which even with the high noise level, can correct the trajectory and increase R from ∼0.3 to 0.8 in each cycle. Although not shown in this
paper, the U-STN1+SPEnKF framework remains stable beyond 10d and shows equally good performance for longer periods of time.
One last point to make here is that within each DA cycle, the maximum forecast accuracy is not when DA occurs but 3–4h later (this is most clearly seen for the case with σ[obs]=1σ[Z] in Fig. 5). A
likely reason behind the further improvement of the performance after DA is the de-noising capability of neural networks when trained on non-noisy training data (Xie et al., 2012).
4.3DDWP+DA with virtual observations: a multi-time-step framework
One might wonder how the performance of the DDWP model (with or without DA) depends on Δt. Figure 6 compares the performance of U-STNx as well as U-NETx for Δt=1, 6, and 12h for 30 random noise-free
initial conditions (no DA). It is clear that the DDWP models with larger Δt outperform the ones with smaller Δt; that is, in terms of forecast accuracy, $\text{U-STN12}>\text{U-STN6}>\text{U-STN1}$.
This trends hold true for both U-STNx and U-NETx, while as discussed before, for the same Δt, the U-STN outperforms U-NET.
This dependence on Δt might seem counterintuitive as it is opposite of what one sees in numerical models, whose forecast errors decrease with smaller time steps. The increase in the forecast errors
of these DDWP models when Δt is decreased is likely due to the non-additive nature of the error accumulation of these autoregressive models. The data-driven models have some degree of generalization
error (for out-of-sample prediction), and every time the model is invoked to predict the next time step, this error is accumulated. For neural networks, this accumulation is not additive and
propagates nonlinearly during the autoregressive prediction. Currently, these error propagations are not understood well enough to build a rigorous framework for estimating the optimal Δt for
data-driven, autoregressive forecasting; however, this behavior has been reported in other studies on nonlinear dynamical systems and can be exploited to formulate multi-time-step data-driven models;
see Liu et al. (2020) for an example (though without DA).
Based on the trends seen in Fig. 6, we propose a novel idea for a multi-time-step DDWP+DA framework, in which the forecasts from the more accurate DDWP with larger Δt are incorporated as virtual
observations, using DA, into the forecasts of the less accurate DDWP with smaller Δt, thus providing overall more accurate short-term forecasts. Figure 7 shows a schematic of this framework for the
case where the U-STN12 model provides the virtual observations that are assimilated using the SPEnKF algorithm in the middle of the 24h DA cycles into the hourly forecasts from U-STN1. At 24th hour,
noisy observations are assimilated using the SPEnKF algorithm as before.
Figure 8 compares the performance of the multi-time-step U-STNx+SPEnKF framework, which uses virtual observations from U-STN12, with that of U-STN1+SPEnKF, which was introduced in Sect. 4.2, for the
case with σ[obs]=0.5σ[Z]. In terms of both RMSE and R, the multi-time-step U-STNx+SPEnKF framework outperforms the U-STN1+SPEnKF framework, as for example, the maximum RMSE of the former is often
comparable to the minimum RMSE of the latter. Figure 9 shows the same analysis but for the case with larger observation noise σ[obs]=σ[Z], which further demonstrates the benefits of the
multi-time-step framework and use of virtual observations.
The multi-time-step framework with assimilated virtual observations introduced here improves the forecasts of short-term intervals by exploiting the non-trivial dependence of the accuracy of
autoregressive, data-driven models on time step size. While hourly forecasts of Z500 may not be necessarily of practical interest, the framework can be applied in general to any state variable and
can be particularly useful for multi-scale systems with a broad range of spatiotemporal scales. A similar idea was used in Bach et al. (2021), wherein data-driven forecasts of oscillatory modes with
singular spectrum analysis and an analog method were used as virtual observations to improve the prediction of a chaotic dynamical system.
In this paper, we propose three novel components for DDWP frameworks to improve their performance: (1) a deep spatial transformer in the latent space to encode the relative spatial relationships of
features of the spatiotemporal data in the network architecture, (2) a stable and inexpensive ensemble-based DA algorithm to ingest noisy observations and correct the forecast trajectory, and (3) a
multi-time-step algorithm, in which the accurate forecasts of a DDWP model that uses a larger time step are assimilated as virtual observations into the less accurate forecasts of a DDWP that uses a
smaller time step, thus improving the accuracy of forecasts at short intervals.
To show the benefits of each component, we use downsampled Z500 data from ERA5 reanalysis and examine the short-term forecast accuracy of the DDWP framework. To summarize the findings, we present the
following points.
1. As show in Sect. 4.1 for noise-free initial conditions (no DA), U-STN12, which uses a deep spatial transformer and Δt=12h, outperforms U-NET12, e.g., extending the average prediction horizon
(when ACC reaches 0.6) from 3.75d (U-NET12) to 5.5d (U-STN12). Examining a few examples of the spatiotemporal evolution of the forecasted Z500 patterns, we can see that U-STN better captures
phenomena such as wave breaking. We further show in Sect. 4.3 based on other metrics that with the same Δt U-STN outperforms U-NET. These results demonstrate the benefits of adding deep spatial
transforms to convolutional networks such as U-NET.
2. As shown in Sect. 4.2, an SPEnKF DA algorithm is coupled with the U-STN1 model. In this framework, the U-STN1 serves as the forward model to generate a large ensemble of forecasts in a
data-driven fashion in each DA cycle (24h), when noisy observations are assimilated. Because U-STN1 is computationally inexpensive, for a state vector of size D, ensembles with 2D=4096 members
are easily generated in each DA cycle, leading to stable, accurate forecasts without the need for localization or inflation of covariance matrices involved in the SPEnKF algorithm. The results
show that DA can be readily coupled with DDWP models when dealing with noisy initial conditions. The results further show that such coupling is substantially facilitated by the fact that large
ensembles can be easily generated with data-driven forward models. Note however that NWP models have a larger number of state variables (O(10^8)) which would make SPEnKF very computationally
expensive; in such cases, further parallelization of the SPEnKF algorithm would be required.
3. As shown in Sect. 4.3, the autoregressive DDWP models (U-STN or U-NET) are more accurate with larger Δt, which is attributed to the nonlinear error accumulation over time. Exploiting this trend
and the ease of coupling DA with DDWP, we show that assimilating the forecasts of U-STN12 into U-STN1+SPEnKF as virtual observations in the middle of the 24h DA cycles can substantially improve
the performance of U-STN1+SPEnKF. These results demonstrate the benefits of the multi-time-step algorithm with virtual observations.
Note that to provide proofs of concept here we have chosen specific parameters, approaches, and setups. However, the framework for adding these three components is extremely flexible, and other
configurations can be easily accommodated. For example, other DA frequencies, Δt, U-NET architectures, or ensemble-based DA algorithms could be used. Furthermore, here we assume that the available
observations are noisy but not sparse. The gain from adding DA to DDWP would be most significant when the observations are noisy and sparse. Moreover, the ability to generate O(1000) ensembles
inexpensively with a DDWP would be particularly beneficial for sparse observations for which the stability of DA is more difficult to achieve without localization and inflation (Asch et al., 2016).
The advantages of the multi-time-step DDWP+DA framework would be most significant when multiple state variables, of different temporal scales, are used, or more importantly, when the DDWP model
consists of several coupled data-driven models for different sets of state variables and processes (Reichstein et al., 2019; Schultz et al., 2021). Moreover, while here we show that ensemble-based DA
algorithms can be inexpensively and stably coupled with DDWP models, variational DA algorithms (Bannister, 2017) could be also used, given that computing the adjoint for the DDWP models can be easily
done using automatic differentiation.
The DDWP models are currently not as accurate as operational NWP models (Weyn et al., 2020; Arcomano et al., 2020; Rasp and Thuerey, 2021; Schultz et al., 2021). However, they can still be useful
through generating large forecast ensembles (Weyn et al., 2021), and there is still much room for improving DDWP frameworks, e.g., using the three components introduced here as well as using transfer
learning, which has been shown recently to work robustly and effectively across a range of problems (e.g., Ham et al., 2019; Chattopadhyay et al., 2020e; Subel et al., 2021; Guan et al., 2022).
Finally, we point out that while here we focus on weather forecasting, the three components can be readily adopted for other parts of the Earth system, such as ocean and land, for which there is a
rapid growth of data and need for forecast and assimilation (e.g., Kumar et al., 2008b, a; Yin et al., 2011; Edwards et al., 2015; Liang et al., 2019).
A1Forecast results with T850 variable
In this section, we have show an example of prediction performance with T850 instead of Z500. In Fig. A1, we can see that U-STN12 shows improved performance as compared to U-NET12 in T850 as well.
A2Comparison with two WeatherBench models
In this section, we present Table A1 to compare U-STN12 model with two WeatherBench models at day 3 and day 5 in terms of RMSE (m^2s^−2) for Z500. Please note that the comparisons made here are with
the U-STN12 without DA and is hence a fair comparison.
AC, MM, and KK designed the study. AC conducted research. AC and PH wrote the article. All authors analyzed and discussed the results. All authors contributed to writing and editing of the article.
The contact author has declared that neither they nor their co-authors have any competing interests.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
We thank Jaideep Pathak, Rambod Mojgani, and Ebrahim Nabizadeh for helpful discussions. We would like to thank Tom Beucler, one anonymous referee and the editor, whose insightful comments,
suggestions, and feedback have greatly improved the clarity of the article. This work was started at National Energy Research Scientific Computing Center (NERSC) as a part of Ashesh Chattopadhyay's
internship in the summer of 2020, under the mentorship of Mustafa Mustafa and Karthik Kashinath, and continued as a part of his PhD work at Rice University, under the supervision of Pedram
Hassanzadeh. This research used resources of NERSC, a U.S. Department of Energy Office of Science User Facility operated under contract no. DE-AC02-05CH11231. Ashesh Chattopadhyay and Pedram
Hassanzadeh were supported by ONR grant N00014-20-1-2722 and NASA grant 80NSSC17K0266. Ashesh Chattopadhyay also thanks the Rice University Ken Kennedy Institute for a BP high-performance computing
(HPC) graduate fellowship. Eviatar Bach was supported by the University of Maryland Flagship Fellowship and Ann G. Wylie Fellowship, as well as by Monsoon Mission II funding (grant
IITMMMIIUNIVMARYLANDUSA2018INT1) provided by the Ministry of Earth Science, Government of India.
This research has been supported by the U.S. Department of Energy, Office of Science (grant no. DE-AC02-05CH11231), the Office of Naval Research (grant no. N00014-20-1-2722), and the National
Aeronautics and Space Administration (grant no. 80NSSC17K0266).
This paper was edited by Xiaomeng Huang and reviewed by Tom Beucler and one anonymous referee.
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Rotational Symmetry Worksheets
Rotational Symmetry Worksheets - Web geometry worksheets for rotations, reflections and symmetry. Scroll down the page for examples and solutions. There are also rotational symmetry worksheets based
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Rotational Symmetry Worksheets
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Worksheet 2 Rotational Symmetry in Cubes and Regular Tetrahedra
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Rotational Symmetry Worksheets 99Worksheets
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rotational symmetry worksheets 4th grade
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rotational symmetry when it is rotated between 0 degrees to 360.
Scroll Down The Page For Examples And Solutions.
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Charles André Barbera, Ph.D. 1980 - Department of Music
Charles André Barbera, Ph.D. 1980
1939-1959 1960-1969 1970-1979 1980-1989 1990-1999 2000-2009 2010-2019 2020-Present
A-C D-F G-I J-L M-O P-R S-V W-Z
Advisor Dissertation Awards
Advisor: Calvin Bower
Dissertation Title: The Persistence of Pythagorean Mathematics in Ancient Musical Thought
Find it in the library here.
Dissertation Abstract:
Several ways of knowing music exist in Western civilization, two of which predominate: grammatical (linguistic) and mathematical. As early as the sixth and fifth centuries B.C., Pythagoras and
Pythagoreans initiated and developed a mathematical way of knowing about the world in general, and in particular about music. Historians of mathematics have long recognized that the Euclidean
generalization of mathematics during the fourth century B.C. rendered obsolete the qualitative, substantive mathematics of the Pythagoreans, which previously had played a participatory role in the
development of Greek mathematics. This generalization, culminating in Euclid’s compilation of the Elements of Geometry (c. 300 B.C.), transformed mathematics into an abstract theory, capable of
accommodating incommensurable magnitudes and generally applicable to all physical sciences.
Historians of music have long recognized that several ancient musical treatises, most of which date from well after the fourth century B.C., contain and rely upon Pythagorean mathematics. My study
investigates why a mathematical way of knowing that was rendered obsolete during the fourth century B.C. by the Euclidean generalization lived on in the musical treatises, persevering for over a
millennium after having been superseded. I conclude that the link and strength between Pythagorean mathematics and ancient musical theory was substantive number. Pythagorean number is as corporeal as
sound, and in this way Pythagorean harmonics (musical theory) distinguishes itself from the incorporeal harmonics of Plato. In addition to mathematical changes and developments, during the fourth
century B.C. Pythagorean musical theory was threatened by the geometrically conceived musical theory of Aristoxenus, but withstood this threat on its own merits. Pythagorean mathematics survived
because Aristoxenus’s Elements of Harmony did not eradicate Pythagorean musical theory. The link between the Pythagorean mathematical and musical theories was of sufficient philosophical strength to
withstand the turn of events during the fourth century B.C.
In this study I present a brief history of Pythagorean mathematics in order to discuss its connection to sound and to music on the bases of: classification, proportional theory, and transfer of
terms. In so doing I define a central tradition for the transmission of Pythagorean mathematics in ancient musical treatises as the corpus of treatises that, in devoting themselves exclusively or
largely to musical matters, exhibit Pythagorean mathematical reasoning. The major mathematical traits and issues occurring in this tradition include: the relation of reason to sensory perception; the
myth of the Pythagorean hammers; the treatment of the semitone; the division of the tetrachord; the arithmetic, geometric, and harmonic means; and the assignment of numbers to notes and ratios to
intervals. This tradition includes the following treatises: Sectio canonis, Nicomachus’s Manual of Harmony, Theon of Smyrna’s Expositio rerum mathematicarum ad legendum Platonem utilium, Gaudentius’s
Introduction to Harmony, and Boethius’s De institutione musica. In addition to these treatises, I discuss works by the following authors: Aristides Quintilianus, Cassiodorus, Censorinus, Chalcidius,
Iamblichus, Macrobius, Martianus Capella, Porphyry, and Proclus. Finally, I use Ptolemy’s Harmonica to evaluate the major mathematical traits and issues found in the musical treatises under
Dr. Barbera was a Professor at St. John’s College in Annapolis, MD. | {"url":"https://music.unc.edu/graduate/phdalumni/phd-alumni-1980-1989/charles-andre-barbera-phd-1980/","timestamp":"2024-11-05T12:56:07Z","content_type":"text/html","content_length":"85141","record_id":"<urn:uuid:deb2c0b3-43e9-4e6d-880c-9d9c6006d3a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00090.warc.gz"} |
Cell References: Relative, Absolute and Mixed Referencing Examples November 6, 2024 - Excel Office
Cell References: Relative, Absolute and Mixed Referencing Examples
Cell references in Excel are very important.
Building a structure or a template in excel using formula one needs to understand the difference between relative, absolute and mixed reference.
Relative Reference
To identify relative referencing, it is simply the ‘cell name’ as illustrated below.
N/B: Cell Name comprises of a column label and a row number of a selected cell.
By default, Excel uses relative references. See the formula in cell D2 below. Cell D2 references (points to) cell B2 and cell C2. Both references are relative.
1. Select cell D2, click on the lower right corner of cell D2 and drag it down to cell D5.
Cell D3 references cell B3 and cell C3. Cell D4 references cell B4 and cell C4. Cell D5 references cell B5 and cell C5. In other words: each cell references its two neighbors on the left.
Absolute Reference
To identify absolute referencing, there is a dollar sign in front of the column label and a dollar sign in front of the row number, e.g $A$1.
See the formula in cell E3 below.
1. To create an absolute reference to cell H3, place a $ symbol in front of the column letter and row number ($H$3) in the formula of cell E3.
2. Now we can quickly drag this formula to the other cells.
The reference to cell H3 is fixed (when we drag the formula down and across). As a result, the correct lengths and widths in inches are calculated.
Mixed Reference
Sometimes we need a combination of relative and absolute reference (mixed reference).
To identify Mixed Reference, there is a dollar sign either in front of the column label or in front of the row number, e.g $A1 or A$1
1. See the formula in cell F2 below.
2. We want to copy this formula to the other cells quickly. Drag cell F2 across one cell, and look at the formula in cell G2.
Do you see what happens? The reference to the price should be a fixed reference to column B. Solution: place a $ symbol in front of the column letter ($B2) in the formula of cell F2. In a similar
way, when we drag cell F2 down, the reference to the reduction should be a fixed reference to row 6. Solution: place a $ symbol in front of the row number (B$6) in the formula of cell F2.
Note: we don’t place a $ symbol in front of the row number of $B2 (this way we allow the reference to change from $B2 (Jeans) to $B3 (Shirts) when we drag the formula down). In a similar way, we
don’t place a $ symbol in front of the column letter of B$6 (this way we allow the reference to change from B$6 (Jan) to C$6 (Feb) and D$6 (Mar) when we drag the formula across).
3. Now we can quickly drag this formula to the other cells.
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Elementary Algebra and Calculus
The book is based on lecture notes Larissa created while teaching large classes of STEM students at a University of widening access and embodies a systematic and efficient teaching method that
marries modern evidence-based pedagogical findings with ideas that can be traced back to such educational and mathematical giants as Socrates and Euler. The courses, which incorporated Larissa's
modules, had been accredited by several UK professional bodies, often after ascertaining that there was no correlation between quality of student degrees and quality of their qualifications on entry. | {"url":"https://bookboon.com/en/elementary-algebra-and-calculus-ebook","timestamp":"2024-11-08T15:30:23Z","content_type":"text/html","content_length":"95235","record_id":"<urn:uuid:6cfa35f4-9b21-45ee-ab25-0614a2a2268e>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00788.warc.gz"} |
What is the SNAP?
The Student Numeracy Assessment and Practice (SNAP) is the Chilliwack district numeracy assessment for all students in grades 2 – 7. It was created by a group of Chilliwack educators and has been
used in all grades 2 – 7 classes since September 2016.
The SNAP is a unique assessment; not only is it a measurement of achievement, but it is intended to be used as a practice tool throughout the entire year. The data it provides should be used to
inform and guide instructional planning. If only used as a summative assessment, the SNAP will not help in achieving one of our main goals, which is to improve students’ proficiency in number sense
and operations.
The SNAP is a two-page assessment that focuses on the foundational skills of mathematics: Number Sense and Operations. It compliments any balanced math program and quickly provides teachers the
information they need for responsive planning and instruction. Access the SNAP Number Sense and Operations templates under the SNAP Templates tab on the website.
SNAP is fully aligned with the BC Curricular Competencies in math. Each area of the assessment is connected to a particular competency, and the competencies are built right into the grading rubric.
Access the grading rubrics under the SNAP Training tab on the website. The rubrics are the same for all grades. It is a good idea to participate in collaborative marking with colleagues to help
establish common expectations.
How to Effectively use the SNAP
SNAP practice does not always need to be on the SNAP templates; in fact, once areas of need are identified, most number sense and operations practice will happen through other strategies, such as
daily high yield number sense routines (e.g. number talks, count around the circle) and whole or small-group instruction. Find resources that support each of the four curricular competencies under
the Resources tab on the website. Explore the Recommended Links for sites that support the teaching and learning of number sense and operations.
Curricular Content and Competency Areas
While the SNAP templates and rubrics are the same for grades 2-7, the curricular content and competency goals (pulled directly from the BC Math Curriculum) change and follow a spiraled approach. The
table below outlines the curricular areas that students will be assessed on at the end of May. The goal is that all students be proficient (3 on the rubric) in their grade-level standards by the end
of the school year. The examples given in the Operations sections are examples of year-end appropriate operations. There are no district-prescribed numbers or operations for the year-end
assessment, but at the request of teachers, numbers and operations have been suggested below to provide guidance.
Remember that the SNAP templates are intended to be used throughout the year for any numbers or operations in your curriculum.
When introducing your students to the SNAP, take your time and explicitly teach and model each component of the assessment. Use content that the students should be confident with from previous
years. You can chunk the assessment into smaller pieces. The Zoom into SNAP templates under the Resources tab on the website chunk the assessment by competency. You can complete SNAPs as a whole
group guided activity and have students work with partners to help build confidence. Have students share their thinking; encourage them to use many different ways to demonstrate their thinking and
Remember that the SNAP templates are intended to be used throughout the year for any numbers or operations in your curriculum.
The SNAP templates
Access templates under SNAP Templates tab.
See Grading Rubrics for specific criteria.
DRAW: The picture must show the value of the number. A written explanation or a legend should be included in the “write to describe your picture” box.
SKIP-COUNTING: Begin at the number and count forwards and backward by numbers chosen by the teacher. *Update – Spring 2024* Teachers have requested guidance on appropriate numbers to use in this
section for the May assessment. We have provided sample numbers based on the curriculum at each grade in the table above.
EQUATIONS: Students who are demonstrating full proficiency will be using grade-appropriate operations in their equations. Teachers should be very specific about their expectations in this section to
avoid students using equations like 4561+1=4562, for example (which is not a grade-appropriate operation in Gr. 4).
REAL-LIFE EXAMPLE: The examples must be realistic and specific. It is important that students demonstrate an understanding of value in their example. For instance, “Wayne Gretzky’s number is 99”
does not show an understanding of value; “we have 99 grade three students in our school” does. Literature and sharing out of real-life examples helps students to make connections to the numbers and
add to their bank of knowledge. There is an excellent list of math picture books on the Coast Metro Elementary Math Project site.
NUMBER LINE: For grades 2-5, the endpoints to the number line are provided. For grades 6 & 7, the students choose their own endpoints according to the number chosen for the assessment. To
demonstrate full proficiency, students will add at least three benchmarks to their number line to help situate the number. Clothesline Math is an excellent routine to help students to become more
proficient with number lines.
REFLECTION: Reflections help increase the value of a learning experience. They allow students to link ideas and construct meaning from their experiences. Students should have opportunities to
reflect on their learning at the end of every lesson. Explicit teaching about how to reflect effectively will improve the quality of student responses in this section; reflection sentence stems are
available in the Connecting and Reflecting Resources page.
See Grading Rubrics for specific criteria.
ESTIMATE: Students will learn to value the skill of estimating through discussions about real-life situations where a person would typically estimate rather than calculate. In which situations would
one prefer a high estimate? A low estimate? Explicit instruction on estimation strategies will allow students to select and use an appropriate strategy for the given operation.
DRAW: Students will visually represent the operation. The visual may or may not contain the solution to the operation. Consider the use of bar diagrams as an appropriate, proportional model for the
operations. Simply replacing the numbers in the operation with a base ten representation does not demonstrate an understanding of the operation.
CALCULATE: Multiple grade-appropriate calculations demonstrate proficient achievement. Students are not required to use the standard algorithm for any operation. Using the reverse operation to
“check” their work is also a recommended strategy. Refer to your grade-specific curriculum elaborations for suggested alternate computation strategies.
REAL-LIFE EXAMPLE OR WORD PROBLEM: Students will provide details on a real-life situation where the given operation would be used to find an amount. Look for evidence that communicates their
understanding of the use of the operation. For example, if the operation was 316-141 a student could suggest, “there were 316 blueberries on the bush and I picked 141 of them.” For the teacher to
know if they understand what the difference between 316 and 141 represents in this situation, they should add, “How many blueberries were left on the bush?”
Grade 2 Math Story: Encourage students to draw pictures to “tell” their story if they do not have the written ability to write a short story. A quick follow up conversation will be required to know
whether students are able to communicate their understanding.
REFLECTION: Reflections help increase the value of a learning experience. They allow students to link ideas and construct meaning from their experiences. Students should have opportunities to
reflect on their learning at the end of every lesson. Explicit teaching about how to reflect effectively will improve the quality of student responses in this section; reflection sentence stems are
available in the Connecting and Reflecting Resources page.
Data Entry
Chilliwack teachers will enter data by the end of November and by the end of May. November data entry is based on the previous year’s outcomes, and is only to be completed by grades 3-7
teachers. For example, grade 4 teachers will assess their students at the beginning of the year based on the grade 3 target outcomes and using the grade 3 templates. All grades 2-7 teachers will
enter data by the end of May based on the current year’s outcomes.
Another unique feature of the SNAP is that students are scored by competency. You will not total or average their scores in the four competencies. Students have until the end of the school year to
practice and become proficient at their grade-level learning standards, however if during your pre-assessments prior to May you have students fully proficient, you may enter their data and create
learning extension opportunities for those students.
The exemplars on the website are intended to represent proficiency in all categories. We will be updating our exemplars on an ongoing basis. Please feel free to send in student samples that you
believe clearly show student proficiency. Scan and send to joanne_britton@sd33.bc.ca.
We are grateful to the dedicated team of Chilliwack educators who crafted and piloted this assessment: Christine Blessin, Jonathan Ferris, Kathy Isaac, Anna Lownie, Shannon McCann, Tammy McKinley,
Kathleen Mitchell, Justin Moore, Kirk Savage, Paul Wojcik | {"url":"https://snap.sd33.bc.ca/node/2","timestamp":"2024-11-05T03:46:50Z","content_type":"text/html","content_length":"31364","record_id":"<urn:uuid:e430b9f0-06a3-41c8-973b-69cad5fc871e>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00350.warc.gz"} |
help with my for loop
My current homework assignment is to code a mortgage calculator that will allow the user to input the appropriate information and return the monthly payment, the interest paid and the principal paid.
An amortization schedule for the entire lenghth of the loan is also required.
I have managed to get my loop counter to step through each month for the entire life of the loan. My problem is that instead of the program also stepping through each payment, it repeats the same
payment information and balance for the complete loan. The balance is never decreased by the amount of each montly payment. I have traced the problem down to something in my loop counter. My problem
is that I cannot find what in my loop counter is wrong. all my math calcuations are correct and the montly payment is accurate. So I believe the problem has to be in my loop. I just need some help
finding it. I have enclosed my code. Please bear in mind that even though i have placed my code in the proper code tags some of the formating may still be off.
// simple mortgage calculator
// programmer - Ronald Scurry
// class - PRG410
// facilitator - Jane Wu
#include <iostream>
#include <cmath> // must include for pow function
using std::cout;
using std::cin;
using std::endl;
int main ()
// defines varibles
double loanAmt = 0.0;
double interest = 0.0;
double term = 0.0;
char quit;
// user input
cout << "Enter interest rate:" << endl;
cin >> interest;
cout << endl;
cout << "Enter term in years:" << endl;
cin >> term;
cout << endl;
cout << "Enter Loan amount:"<< endl;
cin >> loanAmt;
cout << endl;
double monthlyPmt = loanAmt*(interest/1200)/(1-pow(1+(interest/1200),-1*term*12)); // amoritization formula
double loanAmnt = loanAmt;
cout << "For a "<<loanAmnt<<" loan your payment is " <<monthlyPmt<<endl<<endl;
double monthlyInt = loanAmt * (interest/1200);
double principalPaid = monthlyPmt - monthlyInt;
double balance = loanAmt-monthlyPmt;
for (double i=0; i<term*12;i++)
double monthlyInt=balance*(interest/1200);
double principalPaid=monthlyPmt-monthlyInt;
if (balance<1.00)balance=0.0;
cout<<" New loan balance for month "<<i+1<<" is "<<balance<<endl<<endl;
cout<<" Your interest paid for this month is " <<monthlyInt<<endl<<endl;
cout<<" Your Principal paid for this month is " <<principalPaid<<endl;
cout<< "\n";
getchar(); // pauses the program and allows you to view the output
// user can continue in a loop or quit
cout<<"If you wish to continue press C then the enter key on your keyboard.\n";
cout<<"If you wish to quit press Q then the enter key on your keyboard.\n";
//user input
while((quit!='q') && (quit!='Q'));
cout<<"Thank you for trying my simple mortgage calculator\n";
>> The balance is never decreased by the amount of each montly payment.
That's because you have not written any code/statement that decreases the variable "double balance". :)
My guess is you need to change as I've done (Look for "CHANGED")
int main ()
// defines varibles
double loanAmt = 0.0;
double interest = 0.0;
double term = 0.0;
char quit;
// user input
cout << "Enter interest rate:" << endl;
cin >> interest;
cout << endl;
cout << "Enter term in years:" << endl;
cin >> term;
cout << endl;
cout << "Enter Loan amount:"<< endl;
cin >> loanAmt;
cout << endl;
double monthlyPmt = loanAmt*(interest/1200)/(1-pow(1+(interest/1200),-1*term*12)); // amoritization formula
double loanAmnt = loanAmt;
cout << "For a "<<loanAmnt<<" loan your payment is " <<monthlyPmt<<endl<<endl;
double monthlyInt = loanAmt * (interest/1200);
double principalPaid = monthlyPmt - monthlyInt;
double balance = loanAmt-monthlyPmt;
for (double i=0; i<term*12;i++)
double monthlyInt=balance*(interest/1200);
double principalPaid=monthlyPmt-monthlyInt;
if (balance<1.00)
balance -= ( monthlyInt + principalPaid ) ;
cout<<" New loan balance for month "<<i+1<<" is "<<balance<<endl<<endl;
cout<<" Your interest paid for this month is " <<monthlyInt<<endl<<endl;
cout<<" Your Principal paid for this month is " <<principalPaid<<endl;
cout<< "\n";
getchar(); // pauses the program and allows you to view the output
// user can continue in a loop or quit
cout<<"If you wish to continue press C then the enter key on your keyboard.\n";
cout<<"If you wish to quit press Q then the enter key on your keyboard.\n";
//user input
while((quit!='q') && (quit!='Q')) ;
cout<<"Thank you for trying my simple mortgage calculator\n";
return 0 ;
hi gaurav arora here!my problem is that i wanna make a code which reads a string and checks whether the string is palindrome or not.i dont know how exactly to code this program. will someone help me.
i would appriciate.
hey buddy,
Before u get a loud bashing from the moderators of the site regarding ethics , do keep in mind that each new problem is supplosed to be as a new post.
So put up the request again in order to get the answers.
>> The balance is never decreased by the amount of each montly payment.
That's because you have not written any code/statement that decreases the variable "double balance". :)
My guess is you need to change as I've done (Look for "CHANGED")
int main ()
// defines varibles
double loanAmt = 0.0;
double interest = 0.0;
double term = 0.0;
char quit;
// user input
cout << "Enter interest rate:" << endl;
cin >> interest;
cout << endl;
cout << "Enter term in years:" << endl;
cin >> term;
cout << endl;
cout << "Enter Loan amount:"<< endl;
cin >> loanAmt;
cout << endl;
double monthlyPmt = loanAmt*(interest/1200)/(1-pow(1+(interest/1200),-1*term*12)); // amoritization formula
double loanAmnt = loanAmt;
cout << "For a "<<loanAmnt<<" loan your payment is " <<monthlyPmt<<endl<<endl;
double monthlyInt = loanAmt * (interest/1200);
double principalPaid = monthlyPmt - monthlyInt;
double balance = loanAmt-monthlyPmt;
for (double i=0; i<term*12;i++)
double monthlyInt=balance*(interest/1200);
double principalPaid=monthlyPmt-monthlyInt;
if (balance<1.00)
balance -= ( monthlyInt + principalPaid ) ;
cout<<" New loan balance for month "<<i+1<<" is "<<balance<<endl<<endl;
cout<<" Your interest paid for this month is " <<monthlyInt<<endl<<endl;
cout<<" Your Principal paid for this month is " <<principalPaid<<endl;
cout<< "\n";
getchar(); // pauses the program and allows you to view the output
// user can continue in a loop or quit
cout<<"If you wish to continue press C then the enter key on your keyboard.\n";
cout<<"If you wish to quit press Q then the enter key on your keyboard.\n";
//user input
while((quit!='q') && (quit!='Q')) ;
cout<<"Thank you for trying my simple mortgage calculator\n";
return 0 ;
I followed your advise. When I re-compiled and ran the code,
the program deincrimented the remaining balance as it should.
I am now looking for an even more elusive gremlin. As the program steps through each montly payment, (I used a 30 year term to test with) the remaining balance gets to zero at around payment 170 and
remains at zero through to the last month. I suspected the problem was with the math inside the loop. I manipulated the math inside the loop and all I got was inacurate payments. This leads me to
belive the problem is not with the math inside the loop after all, but rather the loop it self.
So I grabed the book and started checking my loop againts what the book says a for loop should be. So far its spot on but still zeros out half way trough the mortgage. So my best guess is either I am
misunderstanding the book, or the problem is something I haven't learned yet. Any help would be apreciated.
Once again thank you for the help you have already provided.
>> balance -= ( monthlyInt + principalPaid )
I'd like you for my banker if you are going to deduct my monthly interest payment from my balance in addition to primical payment!
>> balance -= ( monthlyInt + principalPaid )
I'd like you for my banker if you are going to deduct my monthly interest payment from my balance in addition to primical payment!
If i am understanding you correctly, my problem should be solved
if I change thal line to read >>balance -= (monthlyPmt)
I will try it and let you know.
Strike # 2!!
Change the name balance to what it really is: the OutstandingPrincipalBalance. Now, which part of the monthly payment goes toward paying off the OutstandingPrincipalBalance and which part of the
monthly payment do you pay the lender for the privilege of borrowing the Principal?
Whether the appropriate correction solves your entire program I haven't the foggiest notion, but it's a step in the right direction.
>> balance -= ( monthlyInt + principalPaid )
I'd like you for my banker if you are going to deduct my monthly interest payment from my balance in addition to primical payment!
If i am understanding you correctly, my problem should be solved
if I change thal line to read >>balance -= (monthlyPmt)
I will try it and let you know.
Strike # 3!!!
Try again maybe you'll hit it out of the park this time.
Just subtract the part of the monthly payment that reduces the outstanding principal balance from balance, not the entire monthly monthly payment!
>> balance -= ( monthlyInt + principalPaid )
I'd like you for my banker if you are going to deduct my monthly interest payment from my balance in addition to primical payment!
Yeah that was stupid.. :mrgreen::mrgreen:
So Banking is not an alternate professions for me.. :)
Anyway, I think "balance -= monthlyPmt ;" is the right thing to do and should solve the problem
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4 Meter to Nanometer Calculator | Calculator Bit
4 Meter to Nanometer Calculator
4 Meter = 3999999999.9999995 Nanometer (nm)
Rounded: Nearest 4 digits
4 Meter is 4000000000 Nanometer (nm)
4 Meter is 4 m
How to Convert Meter to Nanometer (Explanation)
• 1 meter = 1000000000 nm (Nearest 4 digits)
• 1 nanometer = 1e-9 m (Nearest 4 digits)
There are 1000000000 Nanometer in 1 Meter. To convert Meter to Nanometer all you need to do is multiple the Meter with 1000000000.
In formula distance is denoted with d
The distance d in Nanometer (nm) is equal to 1000000000 times the distance in meter (m):
L [(nm)] = L [(m)] × 1000000000
Formula for 4 Meter (m) to Nanometer (nm) conversion:
L [(nm)] = 4 m × 1000000000 => 4000000000 nm
How many Nanometer in a Meter
One Meter is equal to 1000000000 Nanometer
1 m = 1 m × 1000000000 => 1000000000 nm
How many Meter in a Nanometer
One Nanometer is equal to 1e-9 Meter
1 nm = 1 nm / 1000000000 => 1e-9 m
The meter (symbol: m) is unit of length in the international System of units (SI), Meter is American spelling and metre is British spelling. Meter was first originally defined in 1793 as 1/10
millionth of the distance from the equator to the North Pole along a great circle. So the length of circle is 40075.017 km. The current definition of meter is described as the length of the path
travelledby light in a vacuum in 1/299792458 of a second, later definition of meter is rephrased to include the definition of a second in terms of the caesium frequency (ΔνCs; 299792458 m/s).
The nanometer (symbol: nm) is unit of length in the International System of Units (SI), equal to 0.000000001 meter or 1x10^-9 meter or 1/1000000000. 1 nanometer is equal to 1000 picometeres. The
nanometer was formerly known as the millimicrometer or millimicron in short. The nanometer is often used to express dimensions on an atomic scale, nanometer is also commonly used to specify the
wavelength of electromagnetic radiation near the visible part of the spectrum.
Cite, Link, or Reference This Page
If you found information page helpful you can cite and reference this page in your work.
• <a href="https://www.calculatorbit.com/en/length/4-meter-to-nanometer">4 Meter to Nanometer Conversion</a>
• "4 Meter to Nanometer Conversion". www.calculatorbit.com. Accessed on November 10 2024. https://www.calculatorbit.com/en/length/4-meter-to-nanometer.
• "4 Meter to Nanometer Conversion". www.calculatorbit.com, https://www.calculatorbit.com/en/length/4-meter-to-nanometer. Accessed 10 November 2024.
• 4 Meter to Nanometer Conversion. www.calculatorbit.com. Retrieved from https://www.calculatorbit.com/en/length/4-meter-to-nanometer.
Meter to Nanometer Calculations Table
Now by following above explained formulas we can prepare a Meter to Nanometer Chart.
Meter (m) Nanometer (nm)
5 5000000000.000001
9 9000000000.000002
Nearest 4 digits
Convert from Meter to other units
Here are some quick links to convert 4 Meter to other length units.
Convert to Meter from other units
Here are some quick links to convert other length units to Meter.
More Meter to Nanometer Calculations
More Nanometer to Meter Calculations
FAQs About Meter and Nanometer
Converting from one Meter to Nanometer or Nanometer to Meter sometimes gets confusing.
Here are some Frequently asked questions answered for you.
Is 1000000000 Nanometer in 1 Meter?
Yes, 1 Meter have 1000000000 (Nearest 4 digits) Nanometer.
What is the symbol for Meter and Nanometer?
Symbol for Meter is m and symbol for Nanometer is nm.
How many Meter makes 1 Nanometer?
1e-9 Meter is euqal to 1 Nanometer.
How many Nanometer in 4 Meter?
Meter have 3999999999.9999995 Nanometer.
How many Nanometer in a Meter?
Meter have 1000000000 (Nearest 4 digits) Nanometer. | {"url":"https://www.calculatorbit.com/en/length/4-meter-to-nanometer","timestamp":"2024-11-10T00:25:59Z","content_type":"text/html","content_length":"52128","record_id":"<urn:uuid:82859211-760e-4d7d-9763-3c96db0acc5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00123.warc.gz"} |
Adding Square Roots Calculator - Free Online Calculator
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Adding Square Roots Calculator
Adding Square Root Calculator helps to find the sum of the given two square roots.
What is Adding Square Roots Calculator?
'Cuemath's Adding Square Roots Calculator' is an online tool that helps to calculate the sum of the given two square roots. Cuemath's online Adding Square Roots Calculator helps you to calculate the
sum of the given two square roots in a few seconds.
Note: Enter numbers up to 3 digits.
How to Use Adding Square Roots Calculator?
Please follow the steps below on how to use the calculator:
• Step 1: Enter the value of 'a' and 'b' in the given input boxes.
• Step 2: Click on "Add" to find the sum of the given two square roots
• Step 3: Click on "Reset" to clear the fields and enter the new values.
How to Add Two Square Roots?
The square root of a number is defined as a number that, when multiplied to itself, gives the product as the original number. The square root of a number 'n' can be written as '√n'. It means that
there is a number 'a' when multiplied again with 'a' gives 'n':
a × a = n. This can also be written as:
a^2 = n or a = √n
Let us try to understand the with the help of an example.
Want to find complex math solutions within seconds?
Use our free online calculator to solve challenging questions. With Cuemath, find solutions in simple and easy steps.
Solved Example:
Add the square roots \(\sqrt[] 25\) and \(\sqrt[] 36\)
Let us write the prime factor of the number and simplify it further.
\(\sqrt[] 25= \sqrt []{5 \times 5} =\sqrt[]{5^2} = 5\)
\(\sqrt[] 36= \sqrt []{6 \times 6} =\sqrt[]{6^2} = 6\)
Add the following square roots \(\sqrt[]25 + \sqrt[] 36\) = 5 + 6 = 11
Similarly, you can try the calculator to find the sum for the following:
• \( \sqrt[] 800\) and \(\sqrt[] {135}\)
• \( \sqrt[] {64}\) and \( \sqrt[] {625}\)
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/calculators/adding-square-roots-calculator/","timestamp":"2024-11-03T07:16:50Z","content_type":"text/html","content_length":"201563","record_id":"<urn:uuid:f51f3bc4-9150-4292-852c-74527230f596>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00295.warc.gz"} |
Key Points In Mathematics For JAMB 2024/2025 Is Out, See Now
This article focused on Key Points In Mathematics For JAMB 2024/2025 – Here are some main focus topics In Mathematics For JAMB 2024 that you have to take note of before JAMB exam starts.
In this article, We shall share lights on some main focus topics that you need to consider in Mathematics for the JAMB 2024/2025. We shall share info about frequently asked Mathematics questions from
the previous years, questions that might be asked and those that mostly come out in Mathematics.
People Also Ask
• What are the topics in Mathematics for Jamb 2024?
• What should I read for Jamb 2024?
• How can I score high in Jamb 2024?
• What are the main topics for Mathematics in Jamb?
Key Points In Mathematics For JAMB 2024/2025
Below are the main key points and topics in Mathematics you should focus on when reading for the JAMB 2024/2025 Examination
1. Number bases
2. Fractions, Decimals, Approximations and Percentages
3. Indices, Logarithms and Surds
4. Sets
5. Polynomials
6. Variation
7. Inequalities
8. Progression
9. Binary Operations
10. Matrices and Determinants
11. Euclidean Geometry
12. Mensuration
13. Loci
14. Coordinate Geometry
15. Trigonometry
16. Differentiation
17. Application of differentiation
18. Integration
19. Representation of data
20. Measures of Location
21. Measures of Dispersion
22. Permutation and Combination
23. Probability
Most Repeated Topics in JAMB Mathematics
Listed below are the most repeated Mathematics topics in JAMB;
1. Angles (Reflex, Obtuse and Acute Angles)
2. Arithmetic and Geometric Progression
3. Bearing
4. Binary Operations
5. Calculus: Differentiation and Integration
6. Circle Theorem
7. Fraction/Evaluation
8. Graphs
9. Indices and Logarithm (Theory questions)
10. Matrices
11. Mensuration (Plane shapes and solid shapes)
12. Partial Fractions
13. Polygons (Interior and exterior angles)
14. Probability (Objective questions)
15. Quadratic equation (Polynomials)
16. Sets and Venn Diagram
17. Standard form
18. Statistics
19. Trigonometry
20. Word problems
Most Repeated Mathematics Questions From Previous JAMB Exams
Below are some of JAMB most repeated Mathematics questions, you’ll answer 40 Mathematics questions in Jamb if it is one of your selected subjects.
1. Correct 241.34 (3 x 10-3)2 to 4 significant figures
A. 0.0014
B. 0.001448
C. 0.0022
D. 0.002172
1. At what rate would a sum of #100.00 deposited for 5 years raise an interest of #7.50?
B. 21/2%
C. 15%
D, 25%
1. Three children shared a basket of mangoes in such a way that the first child took % of the mangoes and the second % of the remainder. What fraction of the mangoes did the third child take?
A. 3/16
B. 7/16
C. 9/16
D. 13/16
1. Simplify and express in standard form (0.00275 x 0.00640/( 0.025 x 0.08)
A. 8.8 x 10-1
B. 8.8 x 102
C. 8.8 x 10-3
D. 8.8 x 103
1. Three brothers in a business deal share the profit at the end of contract. The first received 1/3 of the profit and the second 2/3 of the remainder. If the third received the remaining #
12.000.00, how much profit did they share? A. #60,000.00
B. #54,000.00
C. #48 000.00
D. #42,000.00
1. Simplify v160r2 + ¥(71r4 + ¥100r3 A. 9r2
B. 12V3r
C. 13r
D. ¥139r
1. Simplify ¥27 + 3/V3 A. 4v3 B. 4/vV3 C. 3v3 D. 3V/4
2. Simplify 3Log6g + Log612 + Log664 Log672 A. 5 B. 7776 C. Log631 D. (7776)6
g. Simplify (4 + 1)-1 X-1 y-1
1. Find the sum of the first twenty terms of the arithmetic progression Log a, Log a2, Log a3
A. log a20
B. log a21
C. log a200
D. log a210
1. A carpenter charges #40.00 per day for himself and #10.00 per day for his assistant. If a fleet of a cars were painted for #2,000.00 and the painter worked 10 days more than his assistant, how
much did the assistant receive
A. #32.00
B. #320.00
1. Find the sum of the first 18 terms of the progression 3, 6, 12……..
A. 3(217 1)
B. 3(218) 1)
C. 3(218 + 1)
D. 3(218 1)
1. The angle of a sector of a circle, radius 10.5cm, is 480. calculate the perimeter of the sector
A. 88cm
B. 25.4cm
C. 25.6cm
D. 29.8cm
14. Find the length of a side of a rhombus whose diagonals are 6cm and 8cm.
C. 4cm
15. Each of the interior angles of a regular polygon is 1400. How many sides has the polygon?
16. A cylindrical pipe, made of metal is 3cm, thick if the internal radius of the pipe is 10cm. Find the volume of metal used in making 3m of the pipe
A. 1531cm3
B. 2071mcm3
C. 15,300T1IcmM3
D. 20,700ncm3
17. If the height of two circular cylinders are in the ratio 2:3 and their base radii are in the ratio g. What is the ratio of their volume?
A. 27:32
B. 27:23
C. 23:32
D. 21:27
18. The locus of a point which moves so that it is equidistant from two intersecting straight lines is the
A. perpendicular bisector of the two lines
B. angle bisector of the two lines
C. bisector of the two lines
D. line parallel to the two lines
19. 4, 16, 30, 20, 10, 14 and 26 are represented ona pie chart. Find the sum of the angles of the sectors representing all numbers equal to or greater than 16.
A. 480
B, 840
C. 920
D. 2760
20. The mean of ten positive numbers is 16. when another number is added, the mean becomes 18. find the eleventh number.
A. 3
B. 16
D. 30
21. Two numbers are removed at random fr the numbers 1,2,3 and 4. what is the probability that the sum of the numbers removed is even?
A. 2/3
22. Find the probability that a number selected at random from 41 to 56 is a multi of 9
A. 1/9
B. 2/15
C. 3/16
D. 7/8
23. Musa borrows #10.00 at 2° per month interest and repays #8.00 after 4 months. However much does he still owe?
A. #10.80
B. #10.67
C. #2.80
C. #2.67
24. If 3 gallons of spirit containing 20% water are added to 5gallons of another spirit containing 15% water, what percentage of th mixture is water?
A. 24/5%
B. 167/8%
C. 181/8%
D. 187/8°
25 What is the product of 27/5 (3)3 and (1/5)?
A. 5
D. 1/25
26. Simplify 2log2/5 log72/125 + logg
B. -1 + 2log3
C. -1 +5log2
D. 1-2log2
27. Acar travels from Calabar to Enugu, a distant of pkm with an average speed of ukm per hour and continues to Benin, a distance of qkm, with an average speed of wkm per hour. Find its average speed
from Calabar to Benin.
A. (p+q)up+wa)
B. uw
C. uw(p+q)/iwptuq)
D. (wptuq)/(u+wa)
Want to score 370+ in JAMB? Click HERE to See How Now
28. If w varies inversely as uv/u + v and is equal to 8 when u = 2 and v= 6, finda relationship between u, v, w.
A. upw = 16(u + t)
B. 16ur = 3w(u + t)
C. upw = 12(u + t)
D. 12upw=u+r
29. If glx = x2 + 3x } find glx + 1) g(x)
B. a(x + 2)
C. (2x + 4)
D. (+4)
30. Factorize m3 m2-m+2
A. (me + 1)(m 2)
B.(m + a)(m + 1)(m + 2)
C. (m+ 1)m +1km 2)
D. (m2 + 2)(m 1)
What Are The Topics In Mathematics For JAMB 2024?
Below are some of the most repeated idioms and words in JAMB MAthematics from previous years that might come out again in JAMB 2024:
• Number bases
• Indices, Logarithm and Surds
• Business Mathematics
• Sets
• Fractions, Decimals, Approximations and Percentages
• Algebra and Trigonometry
• Differentiation and Integration (Calculus)
• Statistics
What Should I Read For JAMB 2024 Mathematics?
In preparing for JAMB 2024, you should focus on reading topics like; Number bases, Fractions, Decimals, Approximations and Percentages, Indices, Logarithms and Surds, Sets, Polynomials etc.
How Can I Score High In JAMB 2024?
Scoring high mark in JAMB examination is very easy but you will have to follow some certain procedures and guides, check about that here by clicking on Here
What Are The Main Topics For Mathematics In JAMB?
Below are the main topics in Mathematics you should focus on when reading for the JAMB 2024/2025 Examination
• Number and Numeration
• Algebra
• Geometry / Trigonometry.
• Calculus
• Statistics
I hope we have been able to answer all your questions about JAMB Mathematics key points, topics and most repeated questions in Joint Admission and Matriculation Examination Mathematics? Kindly share
this article now and subscribe to our newsletter for more updates.
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Boost Math Learning with Base Ten Block | LitART Strategies
Using Base Ten Blocks to Boost Math Learning
Base ten blocks are the featured manipulative in Litamatics Math Theme 2: Number and Operations in Base Ten. At LitART, we use actual base ten blocks so learners can enjoy the tactile experience of
physically manipulating the ones, tens, and hundreds, but you can try a sample Litamatics Math activity using the virtual version of base ten blocks found below.
Grade 3-6: Roll, Record, Represent, and Round
Let’s play Roll, Record, Represent, and Round. Try to create the highest score possible on each turn. Whoever creates and represents the highest score wins the round.
Here’s how to play. Roll three ten-sided dice. Write the result as a three-digit number. For example, if you roll a 3, 6, and 9, you could choose from among the results 369, 396, 639, 693, 963, or
936. Choose the number that gives you the highest score and write it on your scorecard.
Allow time. Observe. Provide support as needed.
The largest number possible was 963. Let’s gather base ten blocks to represent the number 963. How many hundreds do you need?
Allow for responses.
You need nine hundreds. How many tens do you need?
Allow for responses.
You need six tens. How many ones do you need?
Allow for responses.
You need three ones. Okay, we did it! Now let’s round our score to the nearest hundred. For example, to round 963 to the nearest hundred check the tens place. If it is 5 or greater, round up to
the next hundred. If it is 4 or lower, leave the hundreds place as-is. What is 963 rounded to the nearest hundred?
Allow for responses.
The result is 1,000 because the number in the tens place is five or greater (and 963 is closer to 1,000 than 900.) Now round 963 to the nearest ten. To round to the nearest ten, look at the digit
in the ones place. If it is 5 or greater, round up.
Allow for responses.
The result is 960 because there is a 3 in the ones place and 3 is less than 5 (and 63 is closer to 60 than to 70.) It is okay if you are not sure how to round yet because we are playing this game
to explore base ten and rounding! Try one on your own before we start the game.
Invite students to roll, record, represent, and round.
Observe. Provide support as needed.
It looks like everyone knows how to play. Compare scores after each player rolls, records, represents, and rounds. Who had the higher score? Do the base ten blocks help you visual the quantities?
Allow for responses.
Now let’s play a few rounds.
Observe. Provide support and clarification as needed.
Ask questions to encourage mathematical thinking and communication.
We all want math learning to be fun, engaging, and effective. Base ten blocks can be used successfully in grades K-8 to explore math concepts throughout the standards including Number and Algebraic
Relationships, Number and Operations in Base Ten, Measurement, Geometry, Fractions, Data, Probability, Problem Solving, and more. Litamatics Math includes base ten blocks, dice, tangrams, craft
sticks, Unifix Cubes, and hundreds of other math manipulatives to engage learners in fun math activities connected to award winning books. | {"url":"https://litart.com/blog/using-base-ten-blocks-to-boost-math-learning/","timestamp":"2024-11-13T20:49:50Z","content_type":"text/html","content_length":"65780","record_id":"<urn:uuid:b48e80d0-f815-4ada-98bf-b9a3b0a2021f>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00735.warc.gz"} |
Talk:Ornstein theory
From Scholarpedia
The article gives a good summary of the Ornstein Theory.
I think one should distinguish between the "Ornstein Theory" and "Ornstein's Theorem". The latter usually refers to his theorem to the effect that entropy completely classifies Bernoulli actions. The
former includes much more such as various criteria for being isomorphic to Bernoulli actions. It was the former that had an enormous impact on ergodic theory lasting to the present and really forms
the subject of the article.
Here are some more specific comments:
1. In the Background the sentence beginning with "We need to add a probability structure..." is confusing. If it refers to the Liouville measure then I don't see the relevance of the "non
reversibility of real phenmenon". If it refers to something else that should be made more explicit.
2. In the last sentence of the Background the phrase "corresponding points are transformed into corresponding points" is obscure. Replacing it with something like "the action of T is taken to the
action of \hat T" would be more informative.
3. In the formula defining the Shannon entropy a minus sign is missing.
4. Matters would be clearer to the general reader if the THEOREM were first stated explicitly in discrete time and then in continuous time. Including the statement as a parenthetical remark in part
b. of the THEOREM in continuous time gives a misleading impression.
5. In the list of consequences of the criteria #6 MIXING should be added to Markov Processes to make the statement correct.
6. At the end of the section on Structural Stability the reference should be to [OW1] not [OW2]
7. There is another characterization of Bernoulli - Almost Block Independence - that perhaps is worth mentioning since it is often used in the theory of random fields.
Second reviewer
This is a very nice presentation, although the going gets tougher towards the end. I second the opinions by the first referee.
The statement of the main theorem is a little awkward - I had a double-take on the first sentence. I suggest: There is an abstract flow \(B_t\) "with the following properties". (Without this, it
sounds like we have a trivial theorem followed by remarks about the fact that there exists a flow.) The statement for the infinite-entropy case looks like it could be made more clear.
The "footnotes" in the "Criteria..." section should probably just be parentheses in the text. By contrast, the first footnote (to a. in the finite-entropy statement) should be part of an introductory
sentence to the section on "The Ornstein Theorem".
The section on structurel stability (i.e., the explanation of statistical stability) is likely unclear to nonexpert readers.
The "join" notation in the section on "Finitely Determined (F.D.)" should be introduced, or there should be a reference to another scholarpedia entry that introduces it. The title "Finitely
Determined (F.D.)" should be completed to something like "Finitely Determined (F.D.) systems". Likewise, the next section title should be something like "very weak Bernoulli (V.W.B.) systems".
"Lebesgue" seems misspelled consistently, and "criterion" (singular) and "criteria" (plural) are mixed up. "ingenuous" should be "ingenious". "more that zero-entropy" should be "more than
zero-entropy". There are numerous other little items to be smoothed, maybe there is a volunteer willing to do this.
Most of my coments on the first version have been addressed but the article
still needs a careful proofreading. Here are a few examples:
1. In the contents "foototes" appears instead of "footnotes. 2. The footnote 7 refers to example 7 and not example 6. 3. Footnote 8 is a very vague reference and probably does not refer to the first
sentence - where it is currently placed. The attribution of what Kolmogorov did and what Rokhlin-Sinai did should be made a little clearer. Finally while there are several transliterations of
Rokhlin' name in Latin characters "Rocklin" is not one of them. 4. In Orbit Equivalence it is worth adding a specific reference to the ORW memoir - since it already appears in the sparse list of
technical sources.
Most of my coments on the first version have been addressed but the article
still needs a careful proofreading. Here are a few examples:
1. In the contents "foototes" appears instead of "footnotes. 2. The footnote 7 refers to example 7 and not example 6. 3. Footnote 8 is a very vague reference and probably does not refer to the first
sentence - where it is currently placed. The attribution of what Kolmogorov did and what Rokhlin-Sinai did should be made a little clearer. Finally while there are several transliterations of
Rokhlin' name in Latin characters "Rocklin" is not one of them. 4. In Orbit Equivalence it is worth adding a specific reference to the ORW memoir - since it already appears in the sparse list of
technical sources. | {"url":"http://var.scholarpedia.org/article/Talk:Ornstein_theory","timestamp":"2024-11-05T07:15:43Z","content_type":"text/html","content_length":"27051","record_id":"<urn:uuid:6513bb23-cf5e-4a88-b6da-f110b20cb3a8>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00537.warc.gz"} |
Calvin Café: The Simons Institute Blog
Today the New York Times has an interview with Cynthia Dwork, who is participating in the ongoing Cryptography program.
Cynthia is well known, among other things, for her work on non-malleability (about which she spoke in the historical papers seminar series) and for her work on differential privacy.
The interview is about concerns about fairness, and you can read about the concept of fairness through awareness here.
Don Knuth on SAT
Next semester, the Simons Institute will host a program on Fine-Grained Complexity and Algorithms Design, one of whose core topics is the exact complexity of satisfiability problems.
Coincidentally, Don Knuth has just posted online a draft of Section 7.2.2.2 of The Art of Computer Programming, which will be part of Volume 4B.
Chapter 7 deals with combinatorial searching; Section 7.2, titled “Generating all possibilities” is about enumeration and exhaustive search; Subsection 7.2.2 is about basic backtracking algorithms;
and Sub-subsection 7.2.2.2 is about SAT.
Even people who are familiar with Don’s famously comprehensive approach to exposition might be surprised to see that this sub-subsection runs to 317 pages, 106 of which are devoted to solutions of
exercises. Unfortunately, there was no space to provide the solution to Exercise 516.
Indistinguishability Obfuscation and Multi-linear Maps: A Brave New World (Guest Post by Ran Canetti)
The following post is written by Ran Canetti
A bunch of us hapless cryptographers got the following boilerplate comment from the FOCS’15 PC:
Overall, submissions related to multi-linear maps and indistinguishability obfuscation were held to a somewhat higher standard. The PC expressed some concern with the recent flurry of activities
pertaining to multi-linear maps and indistinguishability obfuscation, given how little we understand and can say and *prove* about the underlying hardness assumptions.
This comment was clearly written with the best of intentions, to explain views expressed at the PC deliberations. And I’m thankful to it – mainly since it made the underlying misconceptions so
explicit that it mandated a response. So, after discussing and commiserating with colleagues here at Simons, and after amusing ourselves with some analogues of above statement (e.g., “results on NP
completeness are held to a higher standard given how little we understand and can say and *prove* about the hardness solving SAT in polynomial time”), I decided to try to write an – obviously
subjective – account for the recent developments in multilinear maps and indistinguishability obfuscation (IO) and why this exciting research should be embraced and highlighted rather than “held to a
somewhat higher standard” — in spite of how little we understand about the underlying assumptions. The account is aimed at the general CS-theorist.
Let me start by giving rough definitions of the concepts involved. An Indistinguishability Obfuscator (IO) is a randomized algorithm O that takes as input a circuit C and outputs a (distribution
over) circuits O(C) with the properties that:
• C and O(C) have the same functionality,
• O(C) is only polynomially larger than C,
• for any two same-size, functionally equivalent circuits C and C’ we have that O(C) ~ O(C’) (i.e., the distributions over strings representing O(C) and O(C’) are computationally
IO has been proposed as a notion of obfuscation in 2000 (Hada, Barak-Goldreich-Impagliazzo-Sahai-Vadhan-Yang). Indeed, it is arguably a clean and appealing notion – in some sense the natural
extension of semantic security of standard encryption to “functionality-preserving encryption of programs”. However, it has been largely viewed as too weak to be of real applicability or interest.
(There were also no candidate polytime IO schemes, but this in my eyes is a secondary point, see below.)
Things changed dramatically in 2013 when Sahai and Waters demonstrated how IO schemes can be ingeniously combined with other rather “mundane” cryptographic constructs to do some amazing things. Since
then dozens of papers came about that extend the SW techniques and apply them to obtain even more amazing things – that by now have transcended crypto and spilled over to other areas. (e.g.: deniable
encryption, succinct delegation, succinct multi-party computation with hardly any interaction, one message succinct witness hiding and witness indistinguishable proofs, hash functions with
random-oracle-like properties, hardness results for PPAD, and many more). In fact, think about a result in your area that assumes that some computation is done inside a black box – most probably IO
can replace that assumption in one way or another…
Still, my (subjective but distinct) feeling is that we are far from understanding the limits and full power of IO. Furthermore, the study of IO has brought with it a whole new toolbox of techniques
that are intriguing in their own right, and teach us about the power and limitations of working with “encrypted computations”.
So far I have not mentioned any candidate constructions of IO – and indeed the above study is arguably valuable as a pure study of this amazing concept, even without any candidate constructions.
(Paraphrasing Levin on quantum computers, one can take the viewpoint that the above is the study of impossibility results for IO…)
However, unlike quantum computers, here we also have candidate constructions. This is where multilinear maps come to play.
Multi-linear maps are this cool new technical tool (or set of tools) that was recently put forth. (The general concept was proposed by Boneh and Silverberg around 2000, and the first candidate
construction of one of the current variants was presented in 2012 by Garg, Gentry and Halevi.) Essentially, a multilinear map scheme is a fully homomorphic encryption scheme where the public key
provides, in addition to the ability to encrypt elements and perform homomorphic operations on ciphertexts, also the ability to partially decrypt ciphertexts under certain restrictions. There are
many incomparable variants of this general paradigm, which differ both in the functionality provided and in the security guarantees. Indeed, variants appear to be closely tied to candidate
constructions. Furthermore, our understanding of what’s possible here has been evolving considerably, with multiple new constructions, attacks, and fixes reported.
Still, the number and variety of applications of multi-linear maps makes it clear that this “family of primitives” is extremely powerful and well worth studying – both at the level of candidate
constructions, at the level of finding the “right” computational abstractions, and at the level of applications. In a sense, we are here back to the 70’s: we are faced with this new set of algebraic
and number theoretic tools, and are struggling to find good ways to use them and abstract them.
Indeed, some of the most powerful applications of multilinear maps are candidate constructions of IO schemes. The first such candidate construction (by Garg, Gentry, Halevi, Raykova, Sahai and Waters
in 2013) came with only heuristic arguments for security; However more rigorous analyses of this and other constructions, based on well-defined formulations of multi-linear map variants, soon
followed suite. Some of these analyses have eventually been “broken” in the sense that we currently don’t have candidate constructions that satisfy the properties they assume. Still, other analyses
do remain valid. Indeed, there are no attacks against the actual basic IO scheme of Garg etal.
The fact that the only current candidate constructions of IO need to assume existence of some variant of multi-linear maps at some point or another may make it seem as it the two concepts are somehow
tied together. However, there is no reason to believe that this is the case. For all we know, multi-linear maps are just the path first uncovered to IO, and other paths may well be found. Similarly,
even if IO turns out to be unobtainable for some reason, the study of multilinear maps and their power will still remain very relevant.
So, to sum up this long-winded account:
• IO is a natural and fascinating computational concept. Studying its consequences (both within and outside cryptography) is a well worth endeavor.
• Studying new candidate constructions of IO and/or new analyses of their security is another well worth endeavor.
• Multilinear maps are an intriguing and powerful set of techniques and tools. Finding better candidate constructions and abstractions is of central importance to cryptography. Finding new cool
uses of these maps is another intriguing challenge.
• The three should be treated as separate (although touching and potentially interleaving) research efforts.
I’d like to thank Guy Rothblum and Vinod Vaikuntanathan for great comments that significantly improved this post.
Historical talks
During the summer program on cryptography, every Monday afternoon there is a talk on the history of a classic paper or series of papers. Last week, Russell Impagliazzo spoke on his work with Steven
Rudich on the impossibility of basing one-way permutations or key agreement on the existence of one-way functions. The week before, Umesh Vazirani spoke on quantum computing, and earlier Ron Rivest
spoke about the origins of modern cryptography.
All talks are recorded, and the recordings are available here.
Tomorrow afternoon, Daniele Micciancio will speak at 3pm on lattice-based cryptography.
Tomorrow is also the first day of the Workshop on the mathematics of modern cryptography. The program starts at 9:30am Pacific time, and all talk are broadcast live, as usual.
Tune in at 9:30am (12:30pm Eastern)
This Summer, most of the theory of cryptography community is in Berkeley to participate in the Simons Institute program on cryptography.
The program started with week-long series of lectures, all available here, which covered tools such as lattice problems, multilinear maps, oblivious RAMs, scrambled circuits, and differential
privacy, and their applications to homomorphic encryption, obfuscation, delegated computations, and multiparty computations.
This week there is a workshop on secure computations, all whose talks are livestreamed, which starts today at 9:30 Pacific Time with a talk by Amit Sahai on obfuscation.
The Big Workshop on Big Data
This week the Simons Institute is hosting a workshop on Information Theory, Learning and Big Data, and the participation was so high that we had to setup an overflow room.
You can watch all talks live at this link.
Throwback Thursday: Emmanuel Candes
On October 14, 2013, Emmanuel Candès gave this wonderful lecture on the effectiveness of convex programming in data science and in the physical sciences. Slides are here.
For Lent, the Simons Institute is giving up not having a blog. | {"url":"https://blog.simons.berkeley.edu/author/luca/","timestamp":"2024-11-02T05:23:47Z","content_type":"text/html","content_length":"67983","record_id":"<urn:uuid:d4f61644-876f-4f1c-b826-a1530e638db4>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00731.warc.gz"} |
Function Adaptor¶
A function adaptor takes a function(or functions) and returns a new function with enhanced capability. Each adaptor has a functional form with a corresponding class with _adaptor appended to it:
template<class... Fs>FunctionAdaptor_adaptor<Fs...> FunctionAdaptor(Fs...);
Both the functional form and the class form can be used to construct the adaptor.
Static Function Adaptor¶
A static function adaptor is a function adaptor that doesn’t have a functional form. It is only a class. It has an additional requirement that the function is DefaultConstructible:
template<class... Fs>class StaticFunctionAdaptor;
A decorator is a function that returns a function adaptor. The function adaptor may be an unspecified or private type.
template<class... Ts>FunctionAdaptor Decorator(Ts...);
Some parts of the documentation provides the meaning(or equivalence) of an expression. Here is a guide of those symbols:
• f, g, fs, gs, p are functions
• x, y, xs, ys are parameters to a function
• T represents some type
• ... are parameter packs and represent varidiac parameters
All the functions are global function objects except where an explicit template parameter is required on older compilers. However, the documentation still shows the traditional signature since it is
much clearer. So instead of writing this:
struct if_f{ template<class IntegralConstant> constexpr auto operator()(IntegralConstant) const;};const constexpr if_f if_ = {};
The direct function signature is written even though it is actually declared like above:
template<class IntegralConstant>constexpr auto if_(IntegralConstant);
Its usage is the same except it has the extra benefit that the function can be directly passed to another function. | {"url":"https://www.boost.org/doc/libs/1_86_0/libs/hof/doc/html/doc/src/definitions.html","timestamp":"2024-11-11T08:29:05Z","content_type":"text/html","content_length":"11738","record_id":"<urn:uuid:0904121a-7d0d-407a-95b1-d569f6a8e049>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00380.warc.gz"} |
In physics, quasiparticles and collective excitations (which are closely related) are emergent phenomena that occur when a microscopically complicated system such as a solid behaves as if it
contained different weakly interacting particles in vacuum. For example, as an electron travels through a semiconductor, its motion is disturbed in a complex way by its interactions with other
electrons and with atomic nuclei. The electron behaves as though it has a different effective mass travelling unperturbed in vacuum. Such an electron is called an electron quasiparticle.[1] In
another example, the aggregate motion of electrons in the valence band of a semiconductor or a hole band in a metal[2] behave as though the material instead contained positively charged
quasiparticles called electron holes. Other quasiparticles or collective excitations include the phonon (a particle derived from the vibrations of atoms in a solid), the plasmons (a particle derived
from plasma oscillation), and many others.
These particles are typically called quasiparticles if they are related to fermions, and called collective excitations if they are related to bosons,[1] although the precise distinction is not
universally agreed upon.[3] Thus, electrons and electron holes (fermions) are typically called quasiparticles, while phonons and plasmons (baryons) are typically called collective excitations.
The quasiparticle concept is important in condensed matter physics because it can simplify the many-body problem in quantum mechanics.
General introduction
Solids are made of only three kinds of particles: electrons, protons, and neutrons. Quasiparticles are none of these; instead, each of them is an emergent phenomenon that occurs inside the solid.
Therefore, while it is quite possible to have a single particle (electron or proton or neutron) floating in space, a quasiparticle can only exist inside interacting many-particle systems (primarily
Motion in a solid is extremely complicated: Each electron and proton is pushed and pulled (by Coulomb's law) by all the other electrons and protons in the solid (which may themselves be in motion).
It is these strong interactions that make it very difficult to predict and understand the behavior of solids (see many-body problem). On the other hand, the motion of a non-interacting classical
particle is relatively simple; it would move in a straight line at constant velocity. This is the motivation for the concept of quasiparticles: The complicated motion of the real particles in a solid
can be mathematically transformed into the much simpler motion of imagined quasiparticles, which behave more like non-interacting particles.
In summary, quasiparticles are a mathematical tool for simplifying the description of solids.
Relation to many-body quantum mechanics
Any system, no matter how complicated, has a ground state along with an infinite series of higher-energy excited states.
The principal motivation for quasiparticles is that it is almost impossible to directly describe every particle in a macroscopic system. For example, a barely-visible (0.1mm) grain of sand contains
around 1017 nuclei and 1018 electrons. Each of these attracts or repels every other by Coulomb's law. In principle, the Schrödinger equation predicts exactly how this system will behave. But the
Schrödinger equation in this case is a partial differential equation (PDE) on a 3×1018-dimensional vector space—one dimension for each coordinate (x,y,z) of each particle. Directly and
straightforwardly trying to solve such a PDE is impossible in practice. Solving a PDE on a 2-dimensional space is typically much harder than solving a PDE on a 1-dimensional space (whether
analytically or numerically); solving a PDE on a 3-dimensional space is significantly harder still; and thus solving a PDE on a 3×1018-dimensional space is quite impossible by straightforward
One simplifying factor is that the system as a whole, like any quantum system, has a ground state and various excited states with higher and higher energy above the ground state. In many contexts,
only the "low-lying" excited states, with energy reasonably close to the ground state, are relevant. This occurs because of the Boltzmann distribution, which implies that very-high-energy thermal
fluctuations are unlikely to occur at any given temperature.
Quasiparticles and collective excitations are a type of low-lying excited state. For example, a crystal at absolute zero is in the ground state, but if one phonon is added to the crystal (in other
words, if the crystal is made to vibrate slightly at a particular frequency) then the crystal is now in a low-lying excited state. The single phonon is called an elementary excitation. More
generally, low-lying excited states may contain any number of elementary excitations (for example, many phonons, along with other quasiparticles and collective excitations).[4]
When the material is characterized as having "several elementary excitations", this statement presupposes that the different excitations can be combined together. In other words, it presupposes that
the excitations can coexist simultaneously and independently. This is never exactly true. For example, a solid with two identical phonons does not have exactly twice the excitation energy of a solid
with just one phonon, because the crystal vibration is slightly anharmonic. However, in many materials, the elementary excitations are very close to being independent. Therefore, as a starting point,
they are treated as free, independent entities, and then corrections are included via interactions between the elementary excitations, such as "phonon-phonon scattering".
Therefore, using quasiparticles / collective excitations, instead of analyzing 1018 particles, one needs to deal with only a handful of somewhat-independent elementary excitations. It is, therefore,
a very effective approach to simplify the many-body problem in quantum mechanics. This approach is not useful for all systems, however: In strongly correlated materials, the elementary excitations
are so far from being independent that it is not even useful as a starting point to treat them as independent.
Distinction between quasiparticles and collective excitations
Usually, an elementary excitation is called a "quasiparticle" if it is a fermion and a "collective excitation" if it is a boson.[1] However, the precise distinction is not universally agreed upon.[3]
There is a difference in the way that quasiparticles and collective excitations are intuitively envisioned.[3] A quasiparticle is usually thought of as being like a dressed particle: it is built
around a real particle at its "core", but the behavior of the particle is affected by the environment. A standard example is the "electron quasiparticle": an electron in a crystal behaves as if it
had an effective mass which differs from its real mass. On the other hand, a collective excitation is usually imagined to be a reflection of the aggregate behavior of the system, with no single real
particle at its "core". A standard example is the phonon, which characterizes the vibrational motion of every atom in the crystal.
However, these two visualizations leave some ambiguity. For example, a magnon in a ferromagnet can be considered in one of two perfectly equivalent ways: (a) as a mobile defect (a misdirected spin)
in a perfect alignment of magnetic moments or (b) as a quantum of a collective spin wave that involves the precession of many spins. In the first case, the magnon is envisioned as a quasiparticle, in
the second case, as a collective excitation. However, both (a) and (b) are equivalent and correct descriptions. As this example shows, the intuitive distinction between a quasiparticle and a
collective excitation is not particularly important or fundamental.
The problems arising from the collective nature of quasiparticles have also been discussed within the philosophy of science, notably in relation to the identity conditions of quasiparticles and
whether they should be considered "real" by the standards of, for example, entity realism.[5][6]
Effect on bulk properties
By investigating the properties of individual quasiparticles, it is possible to obtain a great deal of information about low-energy systems, including the flow properties and heat capacity.
In the heat capacity example, a crystal can store energy by forming phonons, and/or forming excitons, and/or forming plasmons, etc. Each of these is a separate contribution to the overall heat
The idea of quasiparticles originated in Lev Landau's theory of Fermi liquids, which was originally invented for studying liquid helium-3. For these systems a strong similarity exists between the
notion of quasiparticle and dressed particles in quantum field theory. The dynamics of Landau's theory is defined by a kinetic equation of the mean-field type. A similar equation, the Vlasov
equation, is valid for a plasma in the so-called plasma approximation. In the plasma approximation, charged particles are considered to be moving in the electromagnetic field collectively generated
by all other particles, and hard collisions between the charged particles are neglected. When a kinetic equation of the mean-field type is a valid first-order description of a system, second-order
corrections determine the entropy production, and generally take the form of a Boltzmann-type collision term, in which figure only "far collisions" between virtual particles. In other words, every
type of mean-field kinetic equation, and in fact every mean-field theory, involves a quasiparticle concept.
Examples of quasiparticles and collective excitations
See also: List of quasiparticles
In solids, an electron quasiparticle is an electron as affected by the other forces and interactions in the solid. The electron quasiparticle has the same charge and spin as a "normal" (elementary
particle) electron, and like a normal electron, it is a fermion. However, its mass can differ substantially from that of a normal electron; see the article effective mass.[1] Its electric field is
also modified, as a result of electric field screening. In many other respects, especially in metals under ordinary conditions, these so-called Landau quasiparticles closely resemble familiar
electrons; as Crommie's "quantum corral" showed, an STM can clearly image their interference upon scattering.
A hole is a quasiparticle consisting of the lack of an electron in a state; it is most commonly used in the context of empty states in the valence band of a semiconductor.[1] A hole has the opposite
charge of an electron.
A phonon is a collective excitation associated with the vibration of atoms in a rigid crystal structure. It is a quantum of a sound wave.
A magnon is a collective excitation[1] associated with the electrons' spin structure in a crystal lattice. It is a quantum of a spin wave.
In materials, a photon quasiparticle is a photon as affected by its interactions with the material. In particular, the photon quasiparticle has a modified relation between wavelength and energy
(dispersion relation), as described by the material's index of refraction. It may also be termed a polariton, especially near a resonance of the material. For example, an exciton-polariton is a
superposition of an exciton and a photon; a phonon-polariton is a superposition of a phonon and a photon.
A plasmon is a collective excitation, which is the quantum of plasma oscillations (wherein all the electrons simultaneously oscillate with respect to all the ions).
A polaron is a quasiparticle which comes about when an electron interacts with the polarization of its surrounding ions.
An exciton is an electron and hole bound together.
A plasmariton is a coupled optical phonon and dressed photon consisting of a plasmon and photon.
More specialized examples
A roton is a collective excitation associated with the rotation of a fluid (often a superfluid). It is a quantum of a vortex.
Composite fermions arise in a two-dimensional system subject to a large magnetic field, most famously those systems that exhibit the fractional quantum Hall effect.[7] These quasiparticles are quite
unlike normal particles in two ways. First, their charge can be less than the electron charge e. In fact, they have been observed with charges of e/3, e/4, e/5, and e/7.[8] Second, they can be
anyons, an exotic type of particle that is neither a fermion nor boson.[9]
Stoner excitations in ferromagnetic metals
Bogoliubov quasiparticles in superconductors. Superconductivity is carried by Cooper pairs—usually described as pairs of electrons—that move through the crystal lattice without resistance. A broken
Cooper pair is called a Bogoliubov quasiparticle.[10] It differs from the conventional quasiparticle in metal because it combines the properties of a negatively charged electron and a positively
charged hole (an electron void). Physical objects like impurity atoms, from which quasiparticles scatter in an ordinary metal, only weakly affect the energy of a Cooper pair in a conventional
superconductor. In conventional superconductors, interference between Bogoliubov quasiparticles is tough for an STM to see. Because of their complex global electronic structures, however, high-Tc
cuprate superconductors are another matter. Thus Davis and his colleagues were able to resolve distinctive patterns of quasiparticle interference in Bi-2212.[11]
A Majorana fermion is a particle which equals its own antiparticle, and can emerge as a quasiparticle in certain superconductors, or in a quantum spin liquid.[12]
Magnetic monopoles arise in condensed matter systems such as spin ice and carry an effective magnetic charge as well as being endowed with other typical quasiparticle properties such as an effective
mass. They may be formed through spin flips in frustrated pyrochlore ferromagnets and interact through a Coulomb potential.
Spinon is represented by quasiparticle produced as a result of electron spin-charge separation, and can form both quantum spin liquid and strongly correlated quantum spin liquid in some minerals like
Angulons can be used to describe the rotation of molecules in solvents. First postulated theoretically in 2015,[14] the existence of the angulon was confirmed in February 2017, after a series of
experiments spanning 20 years. Heavy and light species of molecules were found to rotate inside superfluid helium droplets, in good agreement with the angulon theory.[15][16]
Type-II Weyl fermions break Lorentz symmetry, the foundation of the special theory of relativity, which cannot be broken by real particles.[17]
A dislon is a quantized field associated with the quantization of the lattice displacement field of a crystal dislocation. It is a quantum of vibration and static strain field of a dislocation line.
See also
List of quasiparticles
Mean field theory
E. Kaxiras, Atomic and Electronic Structure of Solids, ISBN 0-521-52339-7, pages 65–69.
Ashcroft and Mermin (1976). Solid State Physics (1st ed.). Holt, Reinhart, and Winston. pp. 299–302. ISBN 978-0030839931.
A guide to Feynman diagrams in the many-body problem, by Richard D. Mattuck, p10. "As we have seen, the quasiparticle consists of the original real, individual particle, plus a cloud of disturbed
neighbors. It behaves very much like an individual particle, except that it has an effective mass and a lifetime. But there also exist other kinds of fictitious particles in many-body systems, i.e.
'collective excitations'. These do not center around individual particles, but instead involve collective, wavelike motion of all the particles in the system simultaneously."
Ohtsu, Motoichi; Kobayashi, Kiyoshi; Kawazoe, Tadashi; Yatsui, Takashi; Naruse, Makoto (2008). Principles of Nanophotonics. CRC Press. p. 205. ISBN 9781584889731.
Gelfert, Axel (2003). "Manipulative success and the unreal". International Studies in the Philosophy of Science. 17 (3): 245–263. CiteSeerX 10.1.1.405.2111. doi:10.1080/0269859032000169451.
B. Falkenburg, Particle Metaphysics (The Frontiers Collection), Berlin: Springer 2007, esp. pp. 243–46
"Physics Today Article".
"Cosmos magazine June 2008". Archived from the original on 9 June 2008.
Goldman, Vladimir J (2007). "Fractional quantum Hall effect: A game of five-halves". Nature Physics. 3 (8): 517. Bibcode:2007NatPh...3..517G. doi:10.1038/nphys681.
"Josephson Junctions". Science and Technology Review. Lawrence Livermore National Laboratory.
J. E. Hoffman; McElroy, K; Lee, DH; Lang, KM; Eisaki, H; Uchida, S; Davis, JC; et al. (2002). "Imaging Quasiparticle Interference in Bi2Sr2CaCu2O8+δ". Science. 297 (5584): 1148–51.arXiv:cond-mat/
0209276. Bibcode:2002Sci...297.1148H. doi:10.1126/science.1072640. PMID 12142440.
Banerjee, A.; Bridges, C. A.; Yan, J.-Q.; et al. (4 April 2016). "Proximate Kitaev quantum spin liquid behaviour in a honeycomb magnet". Nature Materials. 15 (7): 733–740.arXiv:1504.08037.
Bibcode:2016NatMa..15..733B. doi:10.1038/nmat4604. PMID 27043779.
Shaginyan, V. R.; et al. (2012). "Identification of Strongly Correlated Spin Liquid in Herbertsmithite". EPL. 97 (5): 56001.arXiv:1111.0179. Bibcode:2012EL.....9756001S. doi:10.1209/0295-5075/97/
Schmidt, Richard; Lemeshko, Mikhail (18 May 2015). "Rotation of Quantum Impurities in the Presence of a Many-Body Environment". Physical Review Letters. 114 (20): 203001.arXiv:1502.03447.
Bibcode:2015PhRvL.114t3001S. doi:10.1103/PhysRevLett.114.203001. PMID 26047225.
Lemeshko, Mikhail (27 February 2017). "Quasiparticle Approach to Molecules Interacting with Quantum Solvents". Physical Review Letters. 118 (9): 095301.arXiv:1610.01604. Bibcode:2017PhRvL.118i5301L.
doi:10.1103/PhysRevLett.118.095301. PMID 28306270.
"Existence of a new quasiparticle demonstrated". Phys.org. Retrieved 1 March 2017.
Xu, S.Y.; Alidoust, N.; Chang, G.; et al. (2 June 2017). "Discovery of Lorentz-violating type II Weyl fermions in LaAlGe". Science Advances. 3 (6): e1603266. Bibcode:2017SciA....3E3266X. doi:10.1126/
sciadv.1603266. PMC 5457030. PMID 28630919.
Li, Mingda; Tsurimaki, Yoichiro; Meng, Qingping; Andrejevic, Nina; Zhu, Yimei; Mahan, Gerald D.; Chen, Gang (2018). "Theory of electron–phonon–dislon interacting system—toward a quantized theory of
dislocations". New Journal of Physics. 20 (2): 023010.arXiv:1708.07143. doi:10.1088/1367-2630/aaa383.
Further reading
L. D. Landau, Soviet Phys. JETP. 3:920 (1957)
L. D. Landau, Soviet Phys. JETP. 5:101 (1957)
A. A. Abrikosov, L. P. Gor'kov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (1963, 1975). Prentice-Hall, New Jersey; Dover Publications, New York.
D. Pines, and P. Nozières, The Theory of Quantum Liquids (1966). W.A. Benjamin, New York. Volume I: Normal Fermi Liquids (1999). Westview Press, Boulder.
J. W. Negele, and H. Orland, Quantum Many-Particle Systems (1998). Westview Press, Boulder
External links
PhysOrg.com – Scientists find new 'quasiparticles'
Curious 'quasiparticles' baffle physicists by Jacqui Hayes, Cosmos 6 June 2008. Accessed June 2008
Particles in physics
Up (quark antiquark) Down (quark antiquark) Charm (quark antiquark) Strange (quark antiquark) Top (quark antiquark) Bottom (quark antiquark)
Electron Positron Muon Antimuon Tau Antitau Electron neutrino Electron antineutrino Muon neutrino Muon antineutrino Tau neutrino Tau antineutrino
Gluino Gravitino Photino
Axino Chargino Higgsino Neutralino Sfermion (Stop squark)
Axion Curvaton Dilaton Dual graviton Graviphoton Graviton Inflaton Leptoquark Magnetic monopole Majoron Majorana fermion Dark photon Planck particle Preon Sterile neutrino Tachyon W′ and Z′ bosons X
and Y bosons
Proton Antiproton Neutron Antineutron Delta baryon Lambda baryon Sigma baryon Xi baryon Omega baryon
Pion Rho meson Eta and eta prime mesons Phi meson J/psi meson Omega meson Upsilon meson Kaon B meson D meson Quarkonium
Exotic hadrons
Tetraquark Pentaquark
Atomic nuclei Atoms Exotic atoms
Positronium Muonium Tauonium Onia Pionium Superatoms Molecules
Hexaquark Heptaquark Skyrmion
Glueball Theta meson T meson
Mesonic molecule Pomeron Diquark R-hadron
Anyon Davydov soliton Dropleton Exciton Hole Magnon Phonon Plasmaron Plasmon Polariton Polaron Roton Trion
Baryons Mesons Particles Quasiparticles Timeline of particle discoveries
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timeline Standard Model
mathematical formulation Subatomic particles Particles Antiparticles Nuclear physics Eightfold way
Quark model Exotic matter Massless particle Relativistic particle Virtual particle Wave–particle duality Particle chauvinism
Wikipedia books
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Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License | {"url":"https://www.hellenicaworld.com/Science/Physics/en/Quasiparticle.html","timestamp":"2024-11-04T07:24:33Z","content_type":"application/xhtml+xml","content_length":"31358","record_id":"<urn:uuid:5cea26bf-2d6f-4614-ba70-ca653da1f197>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00283.warc.gz"} |
Pullback Coherent States, Squeezed States and Quantization
SIGMA 18 (2022), 028, 14 pages arXiv:2108.08082 https://doi.org/10.3842/SIGMA.2022.028
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan
Pullback Coherent States, Squeezed States and Quantization
Rukmini Dey and Kohinoor Ghosh
International Center for Theoretical Sciences, Sivakote, Bangalore, 560089, India
Received December 07, 2021, in final form March 30, 2022; Published online April 09, 2022
In this semi-expository paper, we define certain Rawnsley-type coherent and squeezed states on an integral Kähler manifold (after possibly removing a set of measure zero) and show that they satisfy
some properties which are akin to maximal likelihood property, reproducing kernel property, generalised resolution of identity property and overcompleteness. This is a generalization of a result by
Spera. Next we define the Rawnsley-type pullback coherent and squeezed states on a smooth compact manifold (after possibly removing a set of measure zero) and show that they satisfy similar
properties. Finally we show a Berezin-type quantization involving certain operators acting on a Hilbert space on a compact smooth totally real embedded submanifold of $U$ of real dimension $n$, where
$U$ is an open set in ${\mathbb C}{\rm P}^n$. Any other submanifold for which the criterion of the identity theorem holds exhibit this type of Berezin quantization. Also this type of quantization
holds for totally real submanifolds of real dimension $n$ of a general homogeneous Kähler manifold of real dimension $2n$ for which Berezin quantization exists. In the appendix we review the Rawnsley
and generalized Perelomov coherent states on ${\mathbb C}{\rm P}^n$ (which is a coadjoint orbit) and the fact that these two types of coherent states coincide.
Key words: coherent states; squeezed states; geometric quantization; Berezin quantization.
pdf (431 kb) tex (20 kb)
1. Berceanu S., Schlichenmaier M., Coherent state embeddings, polar divisors and Cauchy formulas, J. Geom. Phys. 34 (2000), 336-358, arXiv:math.DG/9903105.
2. Berezin F.A., Quantization, Math. USSR Izv. 8 (1974), 1109-1165.
3. Boggess A., CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991.
4. Doyle P.H., Hocking J.G., A decomposition theorem for $n$-dimensional manifolds, Proc. Amer. Math. Soc. 13 (1962), 469-471.
5. Kirwin W.D., Coherent states in geometric quantization, J. Geom. Phys. 57 (2007), 531-548, arXiv:math.SG/0502026.
6. Klauder J.R., Skagerstam B.S. (Editors), Coherent states: applications in physics and mathematical physics, World Scientific Publishing Co., Singapore, 1985.
7. Kostant B., Orbits and quantization theory, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, 1971, 395-400.
8. Nair V.P., Quantum field theory: a modern perspective, Graduate Texts in Contemporary Physics, Springer, New York, 2005.
9. Nakahara M., Geometry, topology and physics, 2nd ed., Graduate Student Series in Physics, Institute of Physics, Bristol, 2003.
10. Odzijewicz A., Coherent states and geometric quantization, Comm. Math. Phys. 150 (1992), 385-413.
11. Perelomov A., Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986.
12. Radcliffe J.M., Some problems of coherent spin states, J. Phys. A: Gen. Phys. 4 (1971), 313-323.
13. Rawnsley J.H., Coherent states and Kähler manifolds, Quart. J. Math. Oxford 28 (1977), 403-415.
14. Schnabel R., Squeezed states of light and their applications in laser interferometers, Phys. Rep. 684 (2017), 1-51, arXiv:1611.03986.
15. Spera M., On Kählerian coherent states, in Geometry, Integrability and Quantization (Varna, 1999), Coral Press Sci. Publ., Sofia, 2000, 241-256.
16. Spera M., On some geometric aspects of coherent states, in Coherent states and their applications, Springer Proc. Phys., Vol. 205, Springer, Cham, 2018, 157-172.
17. Woodhouse N., Geometric quantization, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1980.
18. Yaffe L.G., Large $N$ limits as classical mechanics, Rev. Modern Phys. 54 (1982), 407-435. | {"url":"https://emis.de/journals/SIGMA/2022/028/","timestamp":"2024-11-07T03:42:48Z","content_type":"text/html","content_length":"9283","record_id":"<urn:uuid:b5e4f380-f244-4bfa-ab9a-5de9ef7acc45>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00715.warc.gz"} |
Stan Reference Manual
5.6 Variable Types vs.Constraints and Sizes
The type information associated with a variable only contains the underlying type and dimensionality of the variable.
Type Information Excludes Sizes
The size associated with a given variable is not part of its data type. For example, declaring a variable using
real a[3];
declares the variable a to be an array. The fact that it was declared to have size 3 is part of its declaration, but not part of its underlying type.
When are Sizes Checked?
Sizes are determined dynamically (at run time) and thus cannot be type-checked statically when the program is compiled. As a result, any conformance error on size will raise a run-time error. For
example, trying to assign an array of size 5 to an array of size 6 will cause a run-time error. Similarly, multiplying an \(N \times M\) by a \(J \times K\) matrix will raise a run-time error if \(M
\neq J\).
Type Information Excludes Constraints
Like sizes, constraints are not treated as part of a variable’s type in Stan when it comes to the compile-time check of operations it may participate in. Anywhere Stan accepts a matrix as an
argument, it will syntactically accept a correlation matrix or covariance matrix or Cholesky factor. Thus a covariance matrix may be assigned to a matrix and vice-versa.
Similarly, a bounded real may be assigned to an unconstrained real and vice-versa.
When are Function Argument Constraints Checked?
For arguments to functions, constraints are sometimes, but not always checked when the function is called. Exclusions include C++ standard library functions. All probability functions and cumulative
distribution functions check that their arguments are appropriate at run time as the function is called.
When are Declared Variable Constraints Checked?
For data variables, constraints are checked after the variable is read from a data file or other source. For transformed data variables, the check is done after the statements in the transformed data
block have executed. Thus it is legal for intermediate values of variables to not satisfy declared constraints.
For parameters, constraints are enforced by the transform applied and do not need to be checked. For transformed parameters, the check is done after the statements in the transformed parameter block
have executed.
For all blocks defining variables (transformed data, transformed parameters, generated quantities), real values are initialized to NaN and integer values are initialized to the smallest legal integer
(i.e., a large absolute value negative number).
For generated quantities, constraints are enforced after the statements in the generated quantities block have executed.
Type Naming Notation
In order to refer to data types, it is convenient to have a way to refer to them. The type naming notation outlined in this section is not part of the Stan programming language, but rather a
convention adopted in this document to enable a concise description of a type.
Because size information is not part of a data type, data types will be written without size information. For instance, real[] is the type of one-dimensional array of reals and matrix is the type of
matrices. The three-dimensional integer array type is written as int[ , ,], indicating the number slots available for indexing. Similarly, vector[ , ] is the type of a two-dimensional array of | {"url":"https://mc-stan.org/docs/2_18/reference-manual/variable-types-vs-constraints-and-sizes.html","timestamp":"2024-11-04T21:13:55Z","content_type":"text/html","content_length":"108418","record_id":"<urn:uuid:5b53b45f-6094-49f3-81bd-1dc2d56e1ab1>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00563.warc.gz"} |
A. Arithmatic Progression (A.P) → A succession of numbers are i... | Filo
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A. Arithmatic Progression (A.P) A succession of numbers are in arithmotio Progless its Preceding किसी के पिलला term is constant. A. AfGenerd Term of an (A.P) [nith ferm of (A.P)] Let a 8 a ar First tirim
8 Commury difforence nth torm of an A.P *.a: = First Term *cd=Gommon Torm As Sum of n Terms of an Arithmetic Progression: \[ * .\left(S_{n}=\pi / 2[2 d+(n-1) d]\right) \] \[ * \cdot S_{n}=n / 2\left
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A. Arithmatic Progression (A.P) A succession of numbers are in arithmotio Progless its Preceding किसी के पिलला term is constant. A. AfGenerd Term of an (A.P) [nith ferm of (A.P)] Let a 8 a ar
Question First tirim 8 Commury difforence nth torm of an A.P *.a: = First Term *cd=Gommon Torm As Sum of n Terms of an Arithmetic Progression: \[ * .\left(S_{n}=\pi / 2[2 d+(n-1) d]\right) \] \[ *
Text \cdot S_{n}=n / 2\left[t_{1}+t_{2}\right] \] st term +1 last Term
Updated On Mar 31, 2023
Topic Application of Derivatives
Subject Mathematics
Class Class 12
Answer Video solution: 1
Upvotes 64
Avg. Video 4 min | {"url":"https://askfilo.com/user-question-answers-mathematics/a-arithmatic-progression-a-p-a-succession-of-numbers-are-in-34373538313036","timestamp":"2024-11-13T21:06:03Z","content_type":"text/html","content_length":"469760","record_id":"<urn:uuid:fdf70428-75ba-4132-a60e-35250ea7b8f8>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00685.warc.gz"} |
Mona Lisa's smile decoded
The subject of centuries of scrutiny and debate, Mona Lisa's famous smile is routinely described as ambiguous. But is it really that hard to read?
Apparently not.
In an unusual trial, close to 100 per cent of people described her expression as unequivocally "happy", researchers revealed on Friday (March 10).
"We really were astonished," neuroscientist Juergen Kornmeier of the University of Freiburg in Germany, who co-authored the study, told AFP.
Kornmeier and a team used what is arguably the most famous artwork in the world in a study of factors that influence how humans judge visual cues such as facial expressions.
Top Posters In This Topic
• 2
• 2
• 1
• 1
Happy is too giddy a term for me to accept with her...
I think her expression is far deeper and more engaging than the unsupportable notion of happy.
i sense that she resides in contentment. Far more stable than happiness... a state not requiring external conditions to be manifest in any particular manner to maintain the state.
Contentment abides where happiness rises and falls... contenment is a tree where happiness is a breeze.
• 3
Lets be honest, the technique is exquisite and personally I love the painting, but she is a bit of a pudding face. I doubt it was a great portrait of the sitter it's highly stylised as all of his
later work was. It's the same basic idea as John the Baptist, St Anne, the Virgin Mary, neat or on the rocks. Old Leo was a highly overrated artist outside of his own particular vision. Which I
happen to be fond of, but I wouldn't class him in the top 10 in hindsight. And his earlier works, before he set his style, are OK to very, very good, but nothing special.
I think the debate about her smile is a bit of an affectation to be honest.
Edited by oldrover
• 1
To me it's clear it's neither happiness nor sadness. It's colitis.
A little less silly, I've always thought of it as a pan am smile, a smile put up just for politeness.
On 10/3/2017 at 8:39 PM, oldrover said:
[...] I think the debate about her smile is a bit of an affectation to be honest.
I agree with you.
• 1
It's that she knows something which we do not, almost a smug look.
• 1
pardon my ignorance, can someone please tell me what is enigmatic about the painting, I ve watched a few short video of it but I am not finding anything exciting about it or the smile. Has anyone
conducted any experiment where the subjects were not aware of the painting and shown and recorded their opinion?
• 1
The painting is full of sacred geometry and symbolic information.
Silent Trinity
Well I live with my wife and standing face to face with her I can never tell what her mood is most of the time, the eternal problem of men trying to understand women.....so to try and decipher an
expression from an old enigmatic painting will be a difficult task I fear.... lol
Frank Merton
3 minutes ago, Silent Trinity said:
Well I live with my wife and standing face to face with her I can never tell what her mood is most of the time, the eternal problem of men trying to understand women.....so to try and decipher an
expression from an old enigmatic painting will be a difficult task I fear.... lol
I suspect men could learn a lesson from that and not be so transparent.
What we see is what was painted and not necessarily the expression on the face of the sitter, i.e. the artists interpretation of what he saw, or the effect he wanted to create. Over analysing
artwork, however good or bad does nothing for anybody, IMO. You either like it or you don't.
• 1
The right side of her face (viewers' perspective) looks a bit happier than the left side. Overall she looks happy. The smile must be forced to an extent, posing for a portrait. It's pretty
common to see even smiling people with lips that point downwards at the ends in a "sad" direction. It's common to see people who don't look like they're happy with happy-curved lips (Winston
Churchill above for example). Mona's smile is obvious though and a smile denotes happiness, so I'm not exactly astonished at the results. Content would be a more accurate description but if the
choices are happy or sad, then happy.
If smug or secretive were on the list I would have chosen one of those over "happy".
• 1
Does it really matter whether she was happy or sad? How do we know DaVInci didn't take some "liberties" with his interpretation of her face or expression? Don't we have anything better to do than
to waste time on a subject that doesn't matter one way or the other?
• 3
She is looking seductive but trying not to for the paining. He feelings however are engaged toward the artists.
6 hours ago, paperdyer said:
Does it really matter whether she was happy or sad? How do we know DaVInci didn't take some "liberties" with his interpretation of her face or expression? Don't we have anything better to do
than to waste time on a subject that doesn't matter one way or the other?
When Leonardo left for France toward the end of his life he took the painting with him. The man who commissioned it never actually got it by the way. Old Leo was a bit dodgy when it came to coming
through on his orders. So it's likely he was fiddling away with it for a period of years after he last saw the sitter's face.
The painting is more a reflection of his artistic vision than a portrait, as I say it's very similar to the rest of his late work. | {"url":"https://www.unexplained-mysteries.com/forum/topic/305060-mona-lisas-smile-decoded","timestamp":"2024-11-05T13:26:43Z","content_type":"text/html","content_length":"238427","record_id":"<urn:uuid:45dcdfdf-1682-4fa4-8419-b96c104c0da9>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00883.warc.gz"} |
Transactions Online
Hidefumi HIRAISHI, Sonoko MORIYAMA, "Excluded Minors for ℚ-Representability in Algebraic Extension" in IEICE TRANSACTIONS on Fundamentals, vol. E102-A, no. 9, pp. 1017-1021, September 2019, doi:
Abstract: While the graph minor theorem by Robertson and Seymour assures that any minor-closed class of graphs can be characterized by a finite list of excluded minors, such a succinct
characterization by excluded minors is not always possible in matroids which are combinatorial abstraction from graphs. The class of matroids representable over a given infinite field is known to
have an infinite number of excluded minors. In this paper, we show that, for any algebraic element x over the rational field ℚ the degree of whose minimal polynomial is 2, there exist infinitely
many ℚ[x]-representable excluded minors of rank 3 for ℚ-representability. This implies that the knowledge that a given matroid is F-representable where F is a larger field than ℚ does
not decrease the difficulty of excluded minors' characterization of ℚ-representability.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E102.A.1017/_p
author={Hidefumi HIRAISHI, Sonoko MORIYAMA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Excluded Minors for ℚ-Representability in Algebraic Extension},
abstract={While the graph minor theorem by Robertson and Seymour assures that any minor-closed class of graphs can be characterized by a finite list of excluded minors, such a succinct
characterization by excluded minors is not always possible in matroids which are combinatorial abstraction from graphs. The class of matroids representable over a given infinite field is known to
have an infinite number of excluded minors. In this paper, we show that, for any algebraic element x over the rational field ℚ the degree of whose minimal polynomial is 2, there exist infinitely
many ℚ[x]-representable excluded minors of rank 3 for ℚ-representability. This implies that the knowledge that a given matroid is F-representable where F is a larger field than ℚ does
not decrease the difficulty of excluded minors' characterization of ℚ-representability.},
TY - JOUR
TI - Excluded Minors for ℚ-Representability in Algebraic Extension
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1017
EP - 1021
AU - Hidefumi HIRAISHI
AU - Sonoko MORIYAMA
PY - 2019
DO - 10.1587/transfun.E102.A.1017
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E102-A
IS - 9
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - September 2019
AB - While the graph minor theorem by Robertson and Seymour assures that any minor-closed class of graphs can be characterized by a finite list of excluded minors, such a succinct characterization by
excluded minors is not always possible in matroids which are combinatorial abstraction from graphs. The class of matroids representable over a given infinite field is known to have an infinite number
of excluded minors. In this paper, we show that, for any algebraic element x over the rational field ℚ the degree of whose minimal polynomial is 2, there exist infinitely many ℚ[x]
-representable excluded minors of rank 3 for ℚ-representability. This implies that the knowledge that a given matroid is F-representable where F is a larger field than ℚ does not decrease
the difficulty of excluded minors' characterization of ℚ-representability.
ER - | {"url":"https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E102.A.1017/_p","timestamp":"2024-11-02T09:29:18Z","content_type":"text/html","content_length":"60902","record_id":"<urn:uuid:b7e64eb5-77f6-4ab3-91ba-5183eef8b1f7>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00632.warc.gz"} |
How to calculate NPV in Excel? - Initial Return
In this tutorial, we explain how to calculate the net present value (NPV) of an investment project using the NPV and XNPV functions in Excel. We’ll show how to compute the NPV without using these
functions as well.
If you prefer to go over this tutorial in a video format or if you’re visiting this page through our YouTube channel to download the Excel template, just click on the “Video tutorial and Excel
template” section in the contents box below.
Step 1: Enter the project’s cash flows
The first step is to enter the project’s investment cost and future cash flows. The most common scenario is that there will be an investment cost at t=0 and the project will start generating cash
inflows from t=1. Of course, some projects might have investment costs spread over time and/or not start generating cash inflows in t=1.
For the purposes of this tutorial, we will focus on an investment project that requires an investment cost of $1,000 (t=0) and that generates cash inflows of $500 for the following three years (t=1,
t=2, and t=3). We have chosen Jan 1, 2023 as the date for t=0, and determined the remaining dates by adding 365 days at a time: Jan 1, 2024 (t=1), Dec 31, 2024 (t=2), Dec 31, 2025 (t=3). Figure 1
shows how we entered the data into Excel.
Figure 1: Investment project cash flows
Step 2: Compute the net present value
A project’s NPV can be calculated by adding up the present values of its cash flows. However, to compute the present values, we need a discount rate. Let’s assume a discount rate of 10%. Then, the
project’s net present value is:
−$1,000 + $500/1.10 + $500/1.10^2 + $500/1.10^3 = −$1,000 + $454.55 + $413.22 + $375.66 = $243.43
We’ll show you three different ways to obtain this result in Excel.
Option 1: Manual calculation
The first option is to compute the present value of each cash flow and add them up as shown in Figure 2 below. For example, the present value of $500 received on 01-Jan-24 is calculated in cell D5 as
And, the net present value in cell G5 is obtained by adding up the present values of cash flows, including the original investment:
Figure 2: Manual calculation
Option 2: Use Excel’s NPV function
The second (and faster) option is to use Excel’s inbuilt NPV function. This is illustrated in Figure 3. As you can see, the function requires you to enter the discount rate (cell B5) first. This is
followed by each of the cash flows (cells D4, E4, and F4), excluding the original investment at t=0. The function gives you the present value of these future cash flows, so we need to manually
consider the initial investment (cell C4) to arrive at the net present value.
=NPV(B5, D4, E4, F4)+C4
Figure 3: Excel’s NPV function
Option 3: Use Excel’s XNPV function
The third and final option is to use the XNPV function. This function requires you to enter the exact date of each cash flow. Then, the result is obtained by specifying the discount rate (cell B5),
the stream of cash flows including the original investment at t=0 (cells C4:F4), and the array of dates (cells C2:F2) as shown in Figure 4.
=XNPV(B5, C4:F4, C2:F2)
Figure 4: Excel’s XNPV function
Of course, all three options we have examined above give the same result, which is $243.43.
Plotting an NPV schedule
Based on the example we’re using in this tutorial, we should be investing in the project we’re examining as it has a positive NPV of $243.43. But, what if we’re unsure about the discount rate we’re
using? We’ve assumed the correct discount rate to be 10%, but the true discount rate could be higher or lower.
What we could do in such a scenario is to plot an NPV schedule (or profile) in Excel to see whether the project yields a positive net present value even at higher discount rates. In fact, we’d call
the discount rate that sets NPV = 0 as the internal rate of return (IRR).
All we need to is to repeat Step 2 for different values of the discount rate as shown in Figure 5. Specifically, we’ve varied the discount rate between 10% and 30% with 1 percentage point intervals,
calculating the NPV for each discount rate. Then, the plot was obtained by following these steps:
1. Go to the Insert tab.
2. In the “charts” section, click on the “line chart” icon.
3. Once the Chart Design tab becomes active, click on Select Data.
4. Enter the NPV values (cells G5:G25) in the “chart data range” field.
5. Finally, for the “axis label range”, choose the range of discount rate values (cells B5:B25) to produce the plot.
This analysis suggests that the project is viable so long as the discount rate is below 23.4%, which is the project’s IRR.
Figure 5
Video tutorial and Excel template
You can download the Excel template we use in this tutorial below. If you’re visiting this page from our YouTube channel, this is the spreadsheet you’re looking for.
What is next?
If you’ve enjoyed this tutorial, you might find the following tutorials useful as well. | {"url":"https://www.initialreturn.com/how-to-calculate-npv-in-excel","timestamp":"2024-11-13T22:24:31Z","content_type":"text/html","content_length":"92308","record_id":"<urn:uuid:a3981574-6d11-4b67-af3f-e84c3c2df089>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00357.warc.gz"} |
Seasonal fluctuations. Seasonal indices. Method of simple averages
The study of periodic (seasonal) fluctuations. Calculation of average seasonal indices by the method of simple averages.
Note that this calculator calculates seasonal indices for monthly data. For quarterly data please use Seasonal Indices for Quarterly Data.
This is the continuation of a theme started in the article Analytical performance indicators.
Here we will talk about average seasonal indices - analytical indicators of time series characterizing the seasonal fluctuations
The seasonal fluctuations are annual, constantly repeating changes of the studied phenomena. During the analysis of the annual dynamics, you obtain the quantitative characteristics, reflecting the
nature of the indicators' changes by months of the annual cycle.
Seasonal fluctuations are described by seasonal indices, which are calculated as a ratio of the indicator's actual value to some theoretical (predicted) level.
Where i - the number of the seasonal cycle (years), j - the season's ordinal (months).
The obtained values are subject to random deviations. That's why these values are averaged out by years, so you get the average seasonal indices for each period of the annual cycle (months)
$I_{sj}=\frac{\sum_{i=1}^n I_{ij}}{n}$
Depending on the nature of the time series changes, you can calculate the formula with different methods.
I'll review the easiest method - the method of simple averages. The method can be used for the time series with no or negligible downward/upward tendencies. In other words, the observed value
fluctuates around a certain constant value.
$Y_{sj}=\frac{\sum_{i=1}^n Y_{ij}}{n}$, average for each season j (months) for all n periods
$Y_{s0}=\frac{\sum_{i=1, j=1}^{i=n, j=m} Y_{ij}}{nm}$, an average for all periods (n) and seasons (m)
January February March April May June July August September October November December
Digits after the decimal point: 1
Seasonal indices
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If A=[12−1−1],B=[ab1−1]&(A+B˙)2= A2+B2, then the values of ... | Filo
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Question Text If , then the values of are
Updated On Apr 10, 2023
Topic Algebra
Subject Mathematics
Class Class 12
Answer Type Video solution: 3
Upvotes 390
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Christmas Shopping
This problem was created by Henry E. Dudeney and published in "Fireside Puzzles" in The Daily Mail on December 24, 1912.
Note: For more information on British currency and coins in circulation at that time, see British Currency.
Christmas Shopping.
Maud and Christine went into a shop, where, through some curious eccentricity, no change was given, and their joint purchases of Christmas presents amounted together to less than five shillings. "I
find," said Maud, "that I shall require no fewer than six current [as of 1912] coins of the realm to pay for what I have bought."
Christine thought a moment and then exclaimed, "By a strange coincidence, I am in exactly the same difficulty!"
"Then we will pay the two bills together." But, to their astonishment, they still required six coins. What is the smallest possible amount of their purchases—both different.
Maud's purchase amount: s d
Christine's purchase amount: s d
1. If you need to enter fractions of a penny, for example 1½d, you can type "1 1/2". Solutions can also be written as decimals if required.
2. You may enter the two amounts in either order. | {"url":"http://allfunandgames.ca/classics/christmasshopping.shtml","timestamp":"2024-11-01T23:32:33Z","content_type":"text/html","content_length":"13376","record_id":"<urn:uuid:aa7798f7-788b-4053-8c00-3e6b940b5bc8>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00708.warc.gz"} |
Is Maximum Likelihood Useful for Representation Learning?Is Maximum Likelihood Useful for Representation Learning?
May 4, 2017
Is Maximum Likelihood Useful for Representation Learning?
A few weeks ago at the DALI Theory of GANs workshop we had a great discussion about what GANs are even useful for. Pretty much everybody agreed that generating random images from a model is not
really our goal. We either want to use GANs to train conditional probabilistic models (like we do for image super-resolution or speech synthesis, or something along those lines), or as a means of
unsupervised representation learning. Indeed, many papers examine the latent space representations that GANs learn.
But the elephant in the room is that nobody really agrees on what unsupervised representation learning really means, and why any GAN variant should be any better or worse at it than others, whether
GANs or VAEs are better for that. So I thought I'd write a post to address this, focussing now on maximum likelihood learning and variational autoencoders, but many of these things holds true for
variants of GANs as well.
Latent variable models for representation learning
A common approach to unsupervised representation learning is via probabilistic latent variable models (LVMs). A latent variable model is essentially a joint distribution $p(x,z)$ over observations
$x_n$ and associated latent variables $z_n$.
In any latent variable model $p(x,z)$ we can use the posterior $p(z\vert x)$ to - perhaps stochastically - map our datapoints $x_n$ to their representation $z_n$. We want this representation to be
useful. The elephant in the room of course is that no-one really agrees on how to properly define, measure, let alone optimise for usefulness in the unsupervised setting. But one thing is certain:
whatever our definition of the usefulness of the representation it depends on the posterior $p(z\vert x)$. As there are several joint models $p(x,z)$ with exactly the same posterior $p(z\vert x)$,
there can be several LVMs whose posterior and hence representation is equally useful.
The maximum likelihood approach to training an LVM $p(x,z)$ is to maximise the log marginal likelihood $\log p(x)$ of observations. Equivalently, we can say maximum likelihood is trying to reduce the
KL divergence $\operatorname{KL}[p_{\mathcal{D}}(x)|p(x)]$ between the true data distribution $p_{\mathcal{D}}$ and the model marginal $p(x)$
The problem with this is that the marginal $p(x)$ and the posterior $p(z\vert x)$ are orthogonal properties of a LVM: any combination of $p(x)$ and $p(z\vert x)$ defines a valid LVM, and vice versa,
any LVM can be uniquely characterised as a $p(x)$, $p(z\vert x)$ pair. This orthogonality is illustrated in the figure below (the shading corresponds to the objective function value):
So here is the dichotomy: the usefulness of representation only depends on the y axis, $p(z\vert x)$, but maximum likelihood is only sensitive to x axis, $p(x)$. Therefore, maximum likelihood,
without additional constraints on the LVM is a perfectly useless objective function for representation learning, irrespective of how you measure the usefulness of $p(z\vert x)$.
Wait, what?
So, why does it work at all? It works because you never (rarely) do maximum likelihood over all possible LVMs, you only do maximum likelihood on a parametric model class $\mathcal{Q}$ of LVMs. So
let's see what happens if we do maximum likelihood with a constraint:
It is the structure of the model class $\mathcal{Q}$ which introduces some sort of coupling between the marginal likelihood $p_\theta(x)$ and the posterior $p_\theta(z\vert x)$. The objective
function pushes you towards the left, but at some point you're squashed towards the boundary of your model class, which may push you up as well. In reality, the dimensionality of $\mathcal{Q}$ might
be orders of magnitude smaller than the space of all LVMs, so this amoeba is much more likely to be some kind of nonlinear manifold. But you get the idea.
This also means that, if you choose your model-class poorly, you might be able to achieve a higher marginal likelihood, yet end up with a less useful representation:
Here, model class $\mathcal{Q}_2$ has an unfortunate shape which means you can achieve a high likelihood with a pretty useless representation.
Can this happen in practice? Sure it can. If you define a variational autoencoder-like model with Gaussian $p_\theta(z) = \mathcal{N}(0,I)$ and arbitrarily powerful $p_\theta(x\vert z)$, something
like this might happen:
Why is this? If $p_\theta(x\vert z)$ is given arbitrary flexibility, it can in fact learn to ignore $z$ completely and always output the data distribution for each $z$: $p_\theta(x\vert z) = p_{\
mathcal{D}}(x)$. Now, your LVM becomes $p(x,z) = p(z)p_{\mathcal{D}}(x)$, which has perfect likelihood, yet the posterior in this model is independent of your data so it is completely useless for
representation learning. Try it, this actually happens. If you make the generator of a VAE too complex, give it lots of modelling power on top of $z$, it will ignore your latent variables as they are
not needed to achieving a good likelihood.
Note on overfitting
A few commenters confused what I talked about here with the topic of overfitting. This is not overfitting. Overfitting is the discrepancy between training error and generalisation/test error.
Overfitting results from the fact that although we would really like to minimise the KL divergence from the true population distribution of the data, in practice we have to estimate that KL
divergence from a finite training dataset. So in essence we end up minimising the KL divergence between the empirical distribution of the training data. But overfitting is a property of how we
optimise the loss function, not a property of the loss function itself.
Consider $p_{\mathcal{D}}$ which appears on my x axis to be the true, population distribution of data - not the empirical distribution of the training data. Consider my x axis to be the negative test
likelihood on an infinitely large held out test/validation set which is never used for training. If we do this, we have abstracted away from overfitting, indeed, we have abstracted away from machine
learning itself: there is no reference to any training dataset anymore, and I'm not even telling you how to find the optimal $\theta$, all I'm saying that models which have higher test likelihood
don't necessarily provide a more useful representation.
Another way to resolve the overfitting confusion is to consider super simple LVMs with binary or discrete $x$ and $z$. If $x$ and $z$ can only take a small, finite number of values jointly, then the
entire joint distribution can be represented by a joint probability table. In this case, it is not unthinkable that we can have a large enough training set that overfitting should not even be an
issue at all. My argument still holds. When I say arbitrarily flexible I don't mean stupidly overparametrised neural network, I mean flexible enough to contain a large portion of all LVMs that are
concievable between $x$ and $z$.
The take home message is that a good likelihood is not - by itself - a sufficient, nor a necessary condition for an LVM to develop useful representations. Indeed, whether or not a maximum likelihood
LVM develops useful representations depends largely on the restrictions you place on your model class. If you let your model class be arbitrarily flexible, you can achieve a perfect likelihood
without learning a representation at all. These observations are independent of how you define the usefulness of the representation, as long as you use the posterior $p(z\vert x)$ as your
In practice, VAE-type deep generative models restrict the model class by fixing $p(x\vert z)$ to be Gaussian with a fixed covariance. This tends to be a useful restriction as it encourages $z$ to
retain information about $x$.
Finally, the same cricitism holds for vanilla GANs as well - at least as long as we interpret of GANs as an LVM and use the posterior for representation learning. From a generative modelling
perspective, GANs are very similar to maximum likelihood, but instead of minimising the KL divergence, they minimise different divergences between $p_{\mathcal{D}}$ and the marginal model $p(x)$,
such as the Jensen-Shannon, reverse-KL or f-divergences. So the same figures still apply, but with the divergence on the x-axis replaced accordingly.
Variational Inference, ALI, BiGANs, InfoGANs
...stay tuned for follow-ups to this post. In the next one, I will talk about how variational learning is different from maximum likelihood. In variational learning instead of the likelihood, we use
the evidence lower bound (ELBO, or - thanks to Dustin Tran - 💪). As ELBO no longer depends on $p(x)$ alone, it changes the picture slightly, maybe even for the better. | {"url":"https://www.inference.vc/maximum-likelihood-for-representation-learning-2/","timestamp":"2024-11-12T05:24:54Z","content_type":"text/html","content_length":"37000","record_id":"<urn:uuid:82086125-794a-4efd-bede-dabe57c31302>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00262.warc.gz"} |
How to Study for AP® Calculus AB
How To Study For The AP® Calculus AB Exam
The AP® Calculus AB exam tends to be one of the more challenging AP® exams, with about 58% of students achieving a score of 3 or above in 2023. That's good for the sixth-lowest mark of all the AP
course offerings that year. However, the right preparation and resources can make the test substantially more manageable. Whether you’re aiming for a 3 to get some college credit or a 5 to stand out
in your college applications, UWorld has you covered! This AP Calculus AB study plan gives you all the essential knowledge and preparation tips you need to crush the exam and attain your target
How To Prepare Effectively for the AP Calculus AB Exam
In this guide, we will help you with every stage of the process. From the moment you sign up for an AP Calculus AB course to exam day, you can take the following steps to prepare and improve your
chances of getting the score you want.
How to prepare for an AP Calculus AB class
It's spring, and you've signed up to take an AP Calculus AB class next year. What now? Here are some things you can do before the school year starts to hit the ground running:
• Get a head start by preparing over the summer before school starts in the fall.
• Review College Board®'s AP Calculus AB course and exam description.
• If possible, talk to your AP Calculus AB teacher about the class expectations. You might also want to discuss the course rigor with your guidance counselor.
• Brush up on algebra and pre-calculus concepts. Specifically, focus on functions and graphs, rational functions, limits, trigonometry, and the unit circle, factoring polynomials, completing the
square, exponent, and logarithm rules, and working with e and natural logarithms.
• Review formulas from geometry, such as the areas of circles, triangles, rectangles, and trapezoids. Volume and surface area of common 3D shapes like cubes, spheres, cylinders, and cones also show
up occasionally, but the College Board will often provide formulas in a question stem when applicable.
• Study Unit 1 concepts and practice UWorld AP Calculus AB questions on limits. Paul’s online math notes are an excellent, free resource for getting started with calculus and reviewing some
pre-calculus topics. Khan Academy is another free resource if your learning style is geared more toward watching videos than reading.
• Identify the ideal learning strategy for you: how do you best absorb information? Reading? Reading while taking notes? Watching videos? Practicing problems? A combination of these? The summer is
a great time to figure this out.
Self-studying for AP Calc?
AP is hard, but we help make really hard stuff easy to understand.
How to pass the AP Calculus AB class
Most of the concepts in AB Calculus stem from two things: the derivative and the integral. Knowing how to differentiate and integrate various kinds of functions is key to doing well in AP Calculus.
Practice with UWorld questions in Topics 2.5 - 2.10 and 3.1 - 3.2 for derivatives and Unit 6 for integrals. Try Units 4 and 8 to apply the derivative and integral to context for more advanced
practice with these concepts.
Below are some AP Calculus AB study tips for passing your exam:
• Keep working on your homework, even if it feels tough at first—that's how you get better. If you make mistakes, it's a great chance to go back and see how to do it right next time.
• Make flashcards for basic derivative and integral rules and review them regularly until you have them down.
• Try getting some extra practice with UWorld MCQs. Focus on working through derivatives and integrals, particularly chain rule and u-substitution. Ensure you've got these basics solid before
moving on to other topics.
• Practice FRQs using the ones the College Board provides from past years’ tests. Analyze the scoring guidelines to understand what the College Board expects. If you have access to AP Classroom, it
is also a good source of FRQs.
• Take a moment to brush up on the early units; by the end of the year, the material from the beginning might be somewhat rusty.
How to do well on the AP Calculus AB exam
If you're aiming for a top score and really want to nail the basics, focus a lot on word problems and different contexts. Concentrate on Units 4, 5, 7, and 8. It's also a good idea to practice Free
Response Questions (FRQs) using past exams from the College Board. They often use similar types of questions, so spotting patterns can really help.
For instance, every test since 2012 (except possibly for 2020, as those FRQs haven’t been released) has included an FRQ that features a table of data and asks questions about approximating
derivatives with average rates of change or estimating integrals using Riemann sums, all within a specific context. Get comfortable with these kinds of questions to boost your confidence for the
Here are AP Calculus AB study tips for improving your score from a 3 to a 4:
Begin your exam preparation at least three months in advance.
How to score a 5 on the AP Calculus AB exam
Want to score a 5 on the AP Calculus AB exam? Here's a comforting fact: you don't need to ace it to earn that top score. You don’t even need to hit 90%—the usual A-grade mark in most US schools. In
fact, you generally need just under two-thirds of the total points available to score a 5. Remember, everyone makes mistakes, especially in a challenging exam like this, and that’s perfectly fine.
According to the College Board, 22.4% of 273,987 test-takers scored a 5 on the 2023 AP Calculus AB exam.
Aiming for perfection isn't the goal; focus on achieving what you know best. If you're stuck, try to rule out wrong answers and make an educated guess instead of spending too much time on it.
It's crucial to know where to focus your study time. Create a systematic AP Calculus AB study plan that allocates enough time for practicing core concepts. For instance, the derivative of inverse
functions usually appears in just one multiple-choice question per test and isn't typically featured in the free-response section. If this formula trips you up, review it with a few UWorld questions,
and maybe make a flashcard to test yourself now and then. But remember, spending time on more important concepts is more beneficial.
AP Calc is hard, but we can help.
We make difficult concepts easy to understand.
Study strategy for scoring a 5 on the AP Calculus AB exam:
Work on Free Response Questions (FRQs)
Aim to begin your prep at least six months in advance. This gives you ample time to cover all the material without feeling rushed.
Work on Free Response Questions (FRQs)
What units are most difficult to learn or require focus due to complexity?
According to the College Board, the 2023 AP Calculus AB test-takers struggled with Unit 5 the most on MCQs. Slope fields tend to be an area where students struggle, but they don't show up very often
on the test. Focus more attention on separating variables and solving differential equations, where algebra, exponent, and logarithm rules will be valuable. Brush up on previous years' math concepts.
Also, pay close attention to the constant of integration, as it may simplify in unexpected ways. Practice is the key here. Try all of UWorld's questions in 7.6-7.7 to see several examples of how
these questions may appear.
One concept that UWorld's AP Calculus team identified that gives AB students a lot of trouble is volume in Unit 8 at the end of the course (8.7-8.12). Know the formulas for the disk and washer
methods for volumes of revolution and when to apply each. Also, strive to understand how to find cross-sectional volume; area formulas for semicircles and other shapes are also important here.
Generally, word problems and contextual questions tend to be difficult for many AP Calculus AB students. Specifically, related rates questions in Unit 4 give students a lot of trouble (4.4-4.5), and
they consistently appear on the exam as both MCQs and FRQs. The key to these questions is organizing information. Pay close attention to the specific quantity the question asks and what information
it provides (formulas, values of quantities or derivatives/rates, etc.). Scrutinize whether a quantity is “increasing” or “decreasing," as this determines whether the derivative is positive or
negative, respectively. This is also where many geometric area and volume formulas come in handy.
Many AP Calculus AB students make a lot of small mechanical errors that add up throughout the exam. Practice is the key to avoiding these mistakes. Complete UWorld AP Calculus AB practice questions,
and note any common mistakes you make. Review your notes and the explanations for those questions, and then practice more UWorld questions on those topics, keeping a careful eye out for your typical
pitfalls. The College Board knows a lot of common errors students make and structures their answer choices accordingly (and so do we!), so just because the answer you arrived at happens to be an
answer choice doesn’t mean it’s correct.
One common source of such errors is u-substitution in Unit 6. This advanced integration technique has many steps and parts to consider, each of which could be the source of a minor mistake. To avoid
such mistakes, recall that differentiation and integration are inverse operations. If you have time, check your answer by differentiating the result of your integration and making sure it matches the
integrand you started with. If it doesn't, you may have made a mistake along the way.
How to self-study for the AP Calculus AB exam
If you plan to self-study for the AP Calculus AB exam without taking an AP course, you may have a few more obstacles and challenges ahead. However, it is definitely doable. The biggest challenge will
be not having a teacher introduce concepts and help you improve. So, your first step is to find what learning style works best for you:
• Do you learn best from watching videos or visual presentations?
Khan Academy is a great place to start. Their videos introduce the concepts at a great pace, and they provide good base-level questions for building your skills. Another popular set of videos is
Professor Leonard on YouTube. He breaks down concepts well and is passionate about learning. However, his videos are lengthy, so you might need to break them up and spend multiple days on each
• Do you learn best from reading a textbook and taking notes?
A good free resource is Paul's Online Math Notes. As a college professor's notes on calculus, they aren’t specifically geared toward AP, but they are nonetheless a good starting point. If your
budget permits, buy AP Calculus AB study material from Amazon or a secondhand bookstore. Any (single-variable) calculus textbook will work but try to find one specifically for AP Calculus so you
know it’s written for high school students and follows the flow of the AP coursework. Otherwise, follow along with the AP Calculus AB Course and Exam Description to ensure that each topic is
necessary for the exam.
• Do you learn best from practice problems?
Subscribe to a question bank. Our UWorld AP Calculus AB QBank is specifically geared towards helping you learn from mistakes with in-depth explanations. Take similar questions as ones you’ve
previously made mistakes to show your improvement in preparation for the AP Calculus AB exam. If you’ve purchased a textbook, you can practice problems inside or practice some problems in Paul’s
Online Math Notes.
Most students learn from a combination of these approaches, so try different things and see what works best for you. Our recommendation would be to incorporate all of them. Here's a general flow you
can use to facilitate your learning process:
1. Watch a video on a topic (Khan, Leonard, or other) and take notes.
1. If you use Khan Academy, the videos are short enough to watch two or three in one session. They are also very well organized for proper pacing.
2. If you use Professor Leonard, the videos are lengthy, so segment them into multiple viewing sessions. He stitches together multiple lectures into one video, so maybe watch until it cuts to
the next lecture (his clothes will be different).
2. If the topic still isn't clear, rewatch parts or the full video or watch a similar video from another source. Sometimes, hearing or seeing a concept presented in multiple ways can help clear up
3. Read a text explanation of the topic (textbook, Paul’s, or other), and add it to your notes.
4. Work practice problems on the topic (UWorld, Khan, textbook, Paul’s, or other).
1. If you use Khan Academy for videos, take their progress checks and quizzes along the way to help cement the ideas. They are generally not AP-level questions but are great when first learning
a topic.
2. If you use UWorld, read through our explanations, especially on questions you answer incorrectly. We also include hyperlinks to general explanations of concepts or alternate/more detailed
solutions, so we encourage you to explore those as well.
5. Review your notes at the end of your study session.
This flow may or may not work for you. Experiment and figure out what elements to incorporate into your study plan. The next section provides tips on creating a study plan that’s right for you.
We make difficult concepts easy to understand.
AP Calculus AB Study Plan
It's the spring semester and it's crunch time! No matter how much time you have left, we've got some handy tips to help you get ready for the AP Calculus AB exam. Here’s a straightforward study plan
you can follow:
• Begin in February to ensure ample time to cover all units.
• Start with a general review of course topics, unit-by-unit at a quicker pace.
• Utilize resources such as textbooks, class notes, and course materials for concept refreshers.
First Week: Review Growth Areas
Second Week: Focus on FRQs
First Week: Review Growth Areas
Kick off your study by tackling a few UWorld AP Calculus AB MCQs in each topic to spot the ones that are harder to you. Dive into the explanations for any questions you miss. If you’re still feeling
shaky, throw in a video or review your notes on those tough topics. Then, circle back with more UWorld questions to see how much you’ve improved!
Second Week: Focus on FRQs
AP Calculus AB Review/Study Materials
Finally, here is a collection of AB Calculus AB study materials you can use to facilitate your exam prep. Most of these links are sprinkled throughout this guide, but we've listed them in one section
for your convenience.
Question Banks and Practice Problems
• UWorld: AP-level MCQs with in-depth explanations that help you learn from mistakes.
• College Board FRQs: FRQs from past exams that provide excellent practice for those sections.
Video Content
• Khan Academy: Great introductory-level videos to learn concepts with skill-building questions.
• Professor Leonard: A set of college lecture videos where he breaks down the concepts and emphasizes the core ideas behind calculus.
Now that you know how to study for AP Calculus AB, it's time to begin your exam prep. Good luck, and happy studying!
AP Calculus AB Related Topics
Master AP Calculus AB FRQs with our step-by-step guide. Discover strategies for tackling free-response questions to maximize your exam score.
Learn the best tactics for AP Calculus AB MCQs. Our tips help you understand question types and solve multiple-choice questions efficiently.
Access the essential AP Calculus AB Formula Sheet—quickly find all the critical formulas in one place to boost your exam preparation.
Enhance your AP Calculus AB preparation with free practice questions. Test your skills with a range of questions covering all exam topics. | {"url":"https://collegeprep.uworld.com/ap-calculus-ab/study-guide-and-materials/","timestamp":"2024-11-13T11:21:44Z","content_type":"text/html","content_length":"511692","record_id":"<urn:uuid:918989c1-4878-4c99-b6d2-4aed7bc46c0b>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00690.warc.gz"} |
Semi-algebraic proofs, IPS lower bounds, and the τ-conjecture: Can a natural number be negative?
We introduce the binary value principle which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and
algebraic complexity. We prove conditional superpolynomial lower bounds on the Ideal Proof System (IPS) refutation size of this instance, based on a well-known hypothesis by Shub and Smale about the
hardness of computing factorials, where IPS is the strong algebraic proof system introduced by Grochow and Pitassi (2018). Conversely, we show that short IPS refutations of this instance bridge the
gap between sufficiently strong algebraic and semi-algebraic proof systems. Our results extend to full-fledged IPS the paradigm introduced in Forbes et al. (2016), whereby lower bounds against
subsystems of IPS were obtained using restricted algebraic circuit lower bounds, and demonstrate that the binary value principle captures the advantage of semi-algebraic over algebraic reasoning, for
sufficiently strong systems. Specifically, we show the following: Conditional IPS lower bounds: The Shub-Smale hypothesis (1995) implies a superpolynomial lower bound on the size of IPS refutations
of the binary value principle over the rationals defined as the unsatisfiable linear equation g'[i=1]^n 2^i-1x[i] = -1, for boolean x[i]'s. Further, the related τ-conjecture (1995) implies a
superpolynomial lower bound on the size of IPS refutations of a variant of the binary value principle over the ring of rational functions. No prior conditional lower bounds were known for IPS or for
apparently much weaker propositional proof systems such as Frege. Algebraic vs. semi-algebraic proofs: Admitting short refutations of the binary value principle is necessary for any algebraic proof
system to fully simulate any known semi-algebraic proof system, and for strong enough algebraic proof systems it is also sufficient. In particular, we introduce a very strong proof system that
simulates all known semi-algebraic proof systems (and most other known concrete propositional proof systems), under the name Cone Proof System (CPS), as a semi-algebraic analogue of the ideal proof
system: CPS establishes the unsatisfiability of collections of polynomial equalities and inequalities over the reals, by representing sum-of-squares proofs (and extensions) as algebraic circuits. We
prove that IPS is polynomially equivalent to CPS iff IPS admits polynomial-size refutations of the binary value principle (for the language of systems of equations that have no 0/1-solutions), over
both g.,Currency sign and g.,.
שפה מקורית אנגלית
כותר פרסום המארח STOC 2020 - Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
עורכים Konstantin Makarychev, Yury Makarychev, Madhur Tulsiani, Gautam Kamath, Julia Chuzhoy
עמודים 54-67
מספר עמודים 14
מסת"ב (אלקטרוני) 9781450369794
מזהי עצם דיגיטלי (DOIs)
סטטוס פרסום פורסם - 8 יוני 2020
פורסם באופן חיצוני כן
אירוע 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020 - Chicago, ארצות הברית
משך הזמן: 22 יוני 2020 → 26 יוני 2020
סדרות פרסומים
שם Proceedings of the Annual ACM Symposium on Theory of Computing
ISSN (מודפס) 0737-8017
כנס 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020
מדינה/אזור ארצות הברית
עיר Chicago
תקופה 22/06/20 → 26/06/20
טביעת אצבע
להלן מוצגים תחומי המחקר של הפרסום 'Semi-algebraic proofs, IPS lower bounds, and the τ-conjecture: Can a natural number be negative?'. יחד הם יוצרים טביעת אצבע ייחודית. | {"url":"https://cris.ariel.ac.il/iw/publications/semi-algebraic-proofs-ips-lower-bounds-and-the-%CF%84-conjecture-can-a","timestamp":"2024-11-07T03:15:57Z","content_type":"text/html","content_length":"67489","record_id":"<urn:uuid:5ca55e72-733e-4869-8d77-f04cba50dde2>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00135.warc.gz"} |
Convert $6,500 per two weeks to Yearly salary | Talent.com
If you make $6,500 per two weeks, your Yearly salary would be $156,000. This result is obtained by multiplying your base salary by the amount of hours, week, and months you work in a year, assuming
you work 38 hours a week. | {"url":"https://au.talent.com/convert?salary=6500&start=biweekly&end=year","timestamp":"2024-11-03T16:08:20Z","content_type":"text/html","content_length":"62347","record_id":"<urn:uuid:76c9fea0-6653-4789-9b67-3f2e6231c7d0>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00824.warc.gz"} |
Multiplication Zones (18-30) - Build Chromium source for Background Independence | eQuantum
Multiplication Zones (18-30)
Multiplication is the form of expression set equal to the inverse function of symmetrical exponentation which stand as multiplicative identity reflects a point across the origin.
The multiplication zones is a symmetric matrix representing the multilinear relationship of a stretching and shearing within the plane of the base unit.
Square Dimensions
The cyclic behaviors of MEC30 are represented by the pure numerical of the 8 × 8 square product positions that sets continues infinitely.
In this one system, represented as an icon, we can see the distribution profile of the prime numbersas well as their products via a chessboard-like model in Fig. 4. This fundamental chewing
• We show the connection in the MEC 30 mathematically and precisely in the table Fig. 13. The organization of this table is based on the well-known idea of Christian Goldbach.
• That every even number from the should be the sum of two prime numbers. From now on we call all pairs of prime numbers without “1”, 2, 3, 5 Goldbach couples.
The MEC 30 transforms this idea from Christian Goldbach into the structure of a numerical double strand, into an opposite link of the MEC 30 scale. (MEC 30 - pdf)
To implement the above octagonal format of MEC30 then this project will use the unique location of .github across the GitHub platform as listed below:
- [user]/.github
- [user]/[user]/.github
- [user]/[user].github.io/.github
- [user]/[the other user's repos]/.github
- [orgs]/.github
- [orgs]/[orgs]/.github
- [orgs]/[orgs].github.io/.github
- [orgs]/[the other organizations's repos]/.github
Since the first member is 30 then the form is initiated by a matrix of 5 x 6 = 30 which has to be transformed first to 6 x 6 = 36 = 6² prior to the above MEC30's square.
A square system of coupled nonlinear equations can be solved iteratively by Newton’s method. This method uses the Jacobian matrix of the system of equations. (Wikipedia)
Each of the nine (9) types express themselves as one of the three (3) subtypes. So from this perspective, there are 27 distinct patterns which are usually denoted by letters.
Mathematically, this type of system requires 27 letters (1-9, 10–90, 100–900). In practice, the last letter, tav (which has the value 400), is used in combination with itself or other letters from
qof (100) onwards to generate numbers from 500 and above. Alternatively, the 22-letter Hebrew numeral set is sometimes extended to 27 by using 5 sofit (final) forms of the Hebrew letters. (Wikipedia)
We found also a useful method called Square of Nine which was developed by WD Gann to analyze stock market behaviour base on astrological pattern.
He designed a new approach to predicting market behavior using several disciplines, including geometry, astrology, astronomy, and ancient mathematics. They say that not long before his death, Gann
developed a unique trading system. However, he preferred not to make his invention public or share it with anyone. (PipBear)
They are used to determine critical points where an asset's momentum is likely to reverse for the equities when paired with additional momentum
Lineage Retracement
Osp(8|4) | 1 | 2 | 3 | 4 | th
π(10) | 2 | 3 | 5 | 7 | 4th
π(19) | 11 | 13 | 17 | 19 | 8th
π(29) | 23 | 29 | - | - | 10th
π(41) | 31 | 37 | 41 | - | 13th 👈
π(59) | 43 | 47 | 53 | 59 | 17th
----------+----+----+----+-----+- ---
π(72) | 61 | 67 | 71 | - | 20th
π(72+11) | 73 | 79 | 83 | - | 23th
π(83+18) | 89 | 97 |101 | - | 26th
π(101+8) |103 |107 |109 | - | 29th
This density will bring the D3-Brane where the lexer is being assigned per MEC30. Base on the its spin as shown in the above picture this lexer is assigned by Id: 33.
In this short review, we have briefly described the structure of exceptional field theories (ExFT’s), which provide a (T)U-duality covariant approach to supergravity. These are based on symmetries of
toroidally reduced supergravity; however are defined on a general background.
• From the point of view of ExFT the toroidal background is a maximally symmetric solution preserving all U-duality symmetries. In this sense the approach is similar to the embedding tensor
technique, which is used to define gauge supergravity in a covariant and supersymmetry invariant form. Although any particular choice of gauging breaks certain amount of supersymmetry, the
formalism itself is completely invariant. Similarly the U-duality covariant approach is transferred to dynamics of branes in both string and M-theory, whose construction has not been covered
• In the text, we described construction of the field content of exceptional field theories from fields of dimensionally reduced 11-dimensional supergravity, and local and global symmetries of the
theories. Various solutions of the section constraint giving Type IIA/B, 11D and lower-dimensional gauged supergravities have been discussed without going deep into technical details. For
readers’ convenience references for the original works are present.
• As a formalism exceptional field theory has found essential number of application, some of which have been described in this review in more details. In particular, we have covered generalized
twist reductions of ExFTs, which reproduce lower-dimensional gauged supergravities, description of non-geometric brane backgrounds and an algorithm for generating deformations of supergravity
backgrounds based on frame change inside DFT. However, many fascinating applications of the DFT and ExFT formalisms have been left aside.
Among these are non-abelian T-dualities in terms of Poisson-Lie transformations inside DFT [100,101]; generating supersymmetric vacua and consistent truncations of supergravity into lower dimensions
[102,103,104] (for review see [105]); compactifications on non-geometric (Calabi-Yau) backgrounds and construction of cosmological models [54,55,106,107]. (U-Dualities in Type II and M-Theory)
The Golden Ratio "symbolically links each new generation to its ancestors, preserving the continuity of relationship as the means for retracing its lineage."
During the last few years of the 12th century, Fibonacci undertook a series of travels around the Mediterranean. At this time, the world’s most prominent mathematicians were Arabs, and he spent much
time studying with them. His work, whose title translates as the Book of Calculation, was extremely influential in that it popularized the use of the Arabic numerals in Europe, thereby
revolutionizing arithmetic and allowing scientific experiment and discovery to progress more quickly. (Famous Mathematicians)
The mathematically significant Fibonacci sequence defines a set of ratios which can be used to determine probable entry and exit points.
Simply stated, the Golden Ratio establishes that the small is to the large as the large is to the whole. This is usually applied to proportions between segments.
• In the special case of a unit segment, the Golden Ratio provides the only way to divide unity in two parts that are in a geometric progression:
• The side of a pentagon-pentagram can clearly be seen as in relation to its diagonal as 1: (√5 +1)/2 or 1:φ, the Golden Section:
• When you draw all the diagonals in the pentagon you end up with the pentagram. The pentagram shows that the Golden Gnomon, and therefore Golden Ratio, are iteratively contained inside the
• There are set of sequence known as Fibonacci retracement. For unknown reasons, these Fibonacci ratios seem to play a role in the stock market, just as they do in nature. The Fibonacci retracement
levels are 0.236, 0.382, 0.618, and 0.786.
□ The key Fibonacci ratio of 61.8% is found by dividing one number in the series by the number that follows it. For example, 21 divided by 34 equals 0.6176, and 55 divided by 89 equals about
□ The 38.2% ratio is discovered by dividing a number in the series by the number located two spots to the right. For instance, 55 divided by 144 equals approximately 0.38194.
□ The 23.6% ratio is found by dividing one number in the series by the number that is three places to the right. For example, 8 divided by 34 equals about 0.23529.
□ The 78.6% level is given by the square root of 61.8%
• While not officially a Fibonacci ratio, 0.5 is also commonly referenced (50% is derived not from the Fibonacci sequence but rather from the idea that on average stocks retrace half their earlier
This study cascade culminating in the Fibonacci digital root sequence (also period-24). (Golden Ratio - Articles)
(√0.618 - 0.618) x 1000 = (0.786 - 0.618) x 1000 = 0.168 x 1000 = 168 = π(1000)
By parsering 168 primes of 1000 id's across π(π(100 x 100)) - 1 = 200 then the (Δ1) would be initiated. As you may guess they will slightly forms the hexagonal patterns.
The Hexagon chart begins with a 0 in the center, surrounded by the numbers 1 through 6. Each additional layer adds 6 more numbers as we move out, and these numbers are arranged into a Hexagon
formation. This is pretty much as far as Gann went in his descriptions. He basically said, “This works, but you have to figure out how.”One method that I’ve found that works well on all these kinds
of charts is plotting planetary longitude values on them, and looking for patterns. On the chart above, each dot represents the location of a particular planet. The red one at the bottom is the Sun,
and up from it is Mars. These are marked on the chart. Notice that the Sun and Mars are connected along a pink line running through the center of the chart. The idea is that when two planets line up
along a similar line, we have a signal event similar to a conjunction in the sky. Any market vibrating to the Hexagon arrangement should show some kind of response to this situation. (Wave59)
We are focusing to MEC30 so we end up this exponentiation by the famous quote from WD Gann himself stating an important changes by certain repetition of 30.
W.D. Gann: “Stocks make important changes in trend every 30, 60, 120, 150, 210, 240, 300, 330, 360 days or degrees from any important top or bottom.”
In line with 168 there is 330 located of 10th layer. Since the base unit of 30 repeats it self on the center then this 11 x 30 = 330 is pushed to the 10 + 1 = 11th layer.
The Interchange Layers
That is, if the powers of 10 all returned with blue spin, or as a series of rainbows, or evenly alternating colors or other non-random results, ***then I'd say prime numbers appear to have a linkage to 10. I may not know what the the linkage is, just that it appears to exist*** _([HexSpin](https://www.hexspin.com/minor-hexagons/))_.
Within these 1000 primes there will be fractions which end up with 168 identities. This will be the same structure as the seven (7) pàrtitions of addition zones.
The first 1000 prime numbers are silently screaming: “Pay attention to us, for we hold the secret to the distribution of all primes!” We heard the call, and with ‘strange coincidences’ leading the
way have discovered compelling evidence that the 1000th prime number, 7919, is the perfectly positioned cornerstone of a mathematical object with highly organized substructures and stunning
reflectional symmetries. (PrimesDemystified)
1st layer:
It has a total of 1000 numbers
Total primes = π(1000) = 168 primes
2nd layer:
It will start by π(168)+1 as the 40th prime
It has 100x100 numbers or π(π(10000)) = 201 primes
Total cum primes = 168 + (201-40) = 168+161 = 329 primes
3rd layer:
Behave reversal to 2nd layer which has a total of 329 primes
The primes will start by π(π(π(1000th prime)))+1 as the 40th prime
This 1000 primes will become 1000 numbers by 1st layer of the next level
Total of all primes = 329 + (329-40) = 329+289 = 618 = 619-1 = 619 primes - Δ1
By the six (6) matrices above it is clearly shows that there is a fascinating connection between prime numbers and the Golden ratio.
There is a fascinating connection between prime numbers and the Golden ratio.
• The Golden ratio is an irrational number, which means that it cannot be expressed as a ratio of two integers. However, it can be approximated by dividing consecutive Fibonacci numbers.
• Additionally, it has been observed that the frequency of prime numbers in certain sequences related to the Golden ratio (such as the continued fraction expansion of the Golden ratio) appears to
be higher than in other sequences.
• Interestingly, the Fibonacci sequence is closely related to prime numbers, as any two consecutive Fibonacci numbers are always coprime.
However, the exact nature of the relationship between primes and the Golden ratio is still an active area of research.
π(1000) = π(Φ x 618) = 168
During this interchange, the two 16-plets will be crossing over and farther apart but they are more likely to stick together and not switch places.
Another fascinating feature of this array is that any even number of–not necessarily contiguous–factors drawn from any one of the 32 angles in this modulo 120 configuration distribute products to 1
(mod 120) or 49 (mod 120), along with the squares.
• We see from the graphic above that the digital roots of the Fibonacci numbers indexed to our domain (Numbers ≌ to {1,7,11,13,17,19,23,29} modulo 30) repeat palindromically every 32 digits (or 4
thirts) consisting of 16 pairs of bilateral 9 sums.
• The digital root sequence of our domain, on the other hand, repeats every 24 digits (or 3 thirts) and possesses 12 pairs of bilateral 9 sums. The entire Prime Root sequence end-to-end covering
360° has 48 pairs of bilateral 9 sums.
• And finally, the Prime Root elements themselves within the Cirque, consisting of 96 elements, has 48 pairs of bilateral sums totaling 360. Essentially, the prime number highway consists of
infinitely telescoping circles …
• Also note, the digital roots of the Prime Root Set as well as the digital roots of Fibonnaci numbers and Lucas numbers (the latter not shown above) indexed to it all sum to 432 (48x9) in 360°
• The sequence involving Fibonacci digital roots repeats every 120°, and has been documented by the author on the On-Line Encyclopedia of Integer Sequences: Digital root of Fibonacci numbers
indexed by natural numbers not divisible by 2, 3 or 5 (A227896).
• The four faces of our pyramid additively cascade 32 four-times triangular numbers (Note that 4 x 32 = 128 = the perimeter of the square base which has an area of 32^2 = 1024 = 2^10).
• These include Fibo1-3 equivalent 112 (rooted in T7 = 28; 28 x 4 = 112), which creates a pyramidion or capstone in our model, and 2112 (rooted in T32 = 528; 528 x 4 = 2112), which is the index
number of the 1000th prime within our domain, and equals the total number of ‘elements’ used to construct the pyramid.
A thirt, in case you’re wondering, is a useful unit of measure when discussing intervals in natural numbers not divisible by 2, 3 or 5. A thirt, equivalent to one rotation around the Prime Spiral
Sieve is like a mile marker on the prime number highway. If we take the Modulo 30 Prime Spiral Sieve and expand it to Modulo 360, we see that there are 12 thirts in one complete circle, or ‘cirque’
as we’ve dubbed it. Each thirt consists of 8 elements. (PrimesDemystified)
1000 x (π(11) + 360) days = 1000 x 365 days = 1000 years
Both 1/89 and 1/109 have the Fibonacci sequence encoded in their decimal expansions illustrates a period-24 palindromic that bring the powers of pi.
When the digital root of perfect squares is sequenced within a modulo 30 x 3 = modulo 90 horizon, beautiful symmetries in the form of period-24 palindromes are revealed, which the author has
documented on the On-Line Encyclopedia of Integer Sequences as Digital root of squares of numbers not divisible by 2, 3 or 5 (A24092):
1, 4, 4, 7, 1, 1, 7, 4, 7, 1, 7, 4, 4, 7, 1, 7, 4, 7, 1, 1, 7, 4, 4, 1
In the matrix pictured below, we list the first 24 elements of our domain, take their squares, calculate the modulo 90 congruence and digital roots of each square, and display the digital root
factorization dyad for each square (and map their collective bilateral 9 sum symmetry). (PrimesDemystified)
Geometrically, a transformation matrix rotates, stretches, or shears the vectors it acts upon. The corresponding eigenvalue is often represented as the multiplying factor.
In the matrix pictured below, we list the first 24 elements of our domain, take their squares, calculate the modulo 90 congruence and digital roots of each square, and display the digital root
factorization dyad for each square (and map their collective bilateral 9 sum symmetry). (PrimesDemystified)
77s Structure
Let's consider a Metaron's Cube as a geometric figure composed of 13 equal circles with lines from the center of each circle extending out to the centers of the other 12 circles.
The 13 circles of the Metatron’s cube can be seen as a diagonal axis projection of a 3-dimensional cube, as 8 corner spheres and 6 face-centered spheres. Two spheres are projected into the center
from a 3-fold symmetry axis. The face-centered points represent an octahedron. Combined these 14 points represent the face-centered cubic lattice cell. (Wikipedia)
If the four pieces are restructured in the form of a rectangle, it appears that the overall area has inexplicably lost one unit! What has happened?
Notice that the divisions in the original square have been done according to some Fibonacci numbers: 5, 8 and 13=5+8; therefore the sides of the transformed rectangle are also Fibonacci numbers
because it has been constructed additively. Now, do you guess how could we correct the dimensions of the initial square so that the above transformation into a rectangle was area-preserving? Yes, as
it could not be another way round, we need to introduce the Golden Ratio! If the pieces of the square are constructed according to Golden proportions, then the area of the resulting rectangle will
coincide with the area of the square. (Phi particle physics)
Φ = 2,10
Δ = 5,7,17
3': 13,18,25,42
2' » 13 to 77, Δ = 64
2' and 3' » 13 to 45, Δ = 32
2" + 5" = 7" = 77
2"=22, 3"=33, 2" + 3" = 5" = 55
16, 18,
21, 23, 25,
28, 30, 32, 34, 36, 38, 40, 42,
45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77
32 + 11×7 = 109 = ((10th)th prime)
The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a consequence of flavor and color combinations and antimatter, the fermions and bosons are
known to have 48 and 13 variations, respectively.[ (Wikipedia)
$True Prime Pairs:
(5,7), (11,13), (17,19)
Prime Loops:
π(10) = 4 (node)
π(100) = 25 (partition)
π(1000) - 29 = 139 (section)
π(10000) - 29th - 29 = 1091 (segment)
π(100000) - 109th - 109 = 8884 (texture)
Sum: 4 + 25 + 139 + 1091 + 8884 = 10143 (object)
| 168 | 618 |
-----+-----+-----+-----+-----+ ---
19¨ | 2 | 3 | 5 | 7 | 4¤ -----> assigned to "id:30" 19¨
-----+-----+-----+-----+-----+ ---
17¨ | 11 | 13 | 17 | 19 | 4¤ -----> assigned to "id:31" |
+-----+-----+-----+-----+ |
{12¨}| 23 | 29 | 2¤ (M & F) -----> assigned to "id:32" |
+-----+-----+-----+ |
11¨ | 31 | 37 | 41 | 3¤ ---> Np(33) assigned to "id:33" -----> 77¨ ✔️
-----+-----+-----+-----+-----+ |
19¨ | 43 | 47 | 53 | 57 | 4¤ -----> assigned to "id:34" |
+-----+-----+-----+-----+ |
{18¨}| 61 | 63 | 71 | 3¤ -----> assigned to "id:35" |
+-----+-----+-----+-----+-----+-----+-----+-----+-----+ ---
43¨ | 73 | 79 | 87 | 89 | 97 | 101 | 103 | 107 | 109 | 9¤ (C1 & C2) 43¨
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+ ---
139¨ |----- 13¨ -----|------ 15¨ ------|------ 15¨ ------|
| 1 2 3 | 4 5 6 | 7 8 9 |
Δ Δ Δ
Mod 30 Mod 60 Mod 90
Both scheme are carrying a correlation between two (2) number of 89 and 109 which provide the bilateral of 12 to the 24 cells of prime hexagon.
Every repository on GitHub.com comes equipped with a section for hosting documentation, called a wiki. You can use your repository’s wiki to share long-form content about your project, such as how to
use it, how you designed it, or its core principles. (GitHub)
7 x π(89) = 7 x 24 = 168 = π(1000)
Finally we found that the loop corresponds to a quadratic polynomial originated from the 4th coupling of MEC30 which is holded by five (5) cells between 13 and 17.
Further observation of this 13 vs 17 phenomenon also introduces a lower bound of Mod 90 to four (4) of possible length scales in the structure of prime recycling.
It appears that the triangulations and magic squares structuring the distribution of all prime numbers involving symmetry groups rotated by the 8-dimensional algorithms.
In sum, we’re positing that Palindromagon + {9/3} Star Polygon = Regular Enneazetton.
• The significance of this ‘chain-of-events’ is that we can state with deterministic certainty that cycling the period-24 digital root dyads of both twin primes and the modulo 90 factorization
sequences of numbers not divisible by 2, 3, or 5 generates an infinite progression of these complex polygons possessing stunning reflectional and translational symmetries.
• Lastly, let’s compare the above-pictured ‘enneazetton’ to an 18-gon 9-point star generated by the first three primes; 2, 3 and 5 (pictured below), and we see that they are identical, save for the
number of sides (9 vs. 18). They are essentially convex and concave versions of each other.
This is geometric confirmation of the deep if not profound connection between the three twin prime distribution channels (which remember have 2, 3, and 5 encoded in their Prime Spiral Sieve angles)
and the first three primes, 2, 3, and 5. (PrimesDemystified)
The symmetries that come into focus when the lense aperature, of the Prime Spiral Sieve is tripled to modulo 90, synchronizing its modulus with its period-24 digital root.
Palindromic Sequence
The terminating digits of the prime root angles (24,264,868; see illustration of Prime Spiral Sieve) when added to their reversal (86,846,242) = 111,111,110, not to mention this sequence possesses
symmetries that dovetail perfectly with the prime root and Fibo sequences.
• And when you combine the terminating digit symmetries described above, capturing three rotations around the sieve in their actual sequences, you produce the ultimate combinatorial symmetry:
• The pattern of 9’s created by decomposing and summing either the digits of Fibonacci numbers indexed to the first two rotations of the spiral (a palindromic pattern {1393717997173931} that
repeats every 16 Fibo index numbers) or, similarly, decomposing and summing the prime root angles.
• The decomposition works as follows (in digit sum arithmetic this would be termed summing to the digital root) of F17 (the 17th Fibonacci number) = 1597 = 1 + 5 + 9 + 7 = 22 = 2 + 2 = 4:Parsing
the squares by their mod 90 congruence reveals that there are 96 perfect squares generated with each 4 * 90 = 360 degree cycle, which distribute 16 squares to each of 6 mod 90 congruence sub-sets
defined as n congruent to {1, 19, 31, 49, 61, 79} forming 4 bilateral 80 sums. (PrimesDemystified)
The vortex theory of the atom was a 19th-century attempt by William Thomson (later Lord Kelvin) to explain why the atoms recently discovered by chemists came in only relatively few varieties but in
very great numbers of each kind. Based on the idea of stable, knotted vortices in the ether or aether, it contributed an important mathematical legacy.
• The vortex theory of the atom was based on the observation that a stable vortex can be created in a fluid by making it into a ring with no ends. Such vortices could be sustained in the
luminiferous aether, a hypothetical fluid thought at the time to pervade all of space. In the vortex theory of the atom, a chemical atom is modelled by such a vortex in the aether.
• Knots can be tied in the core of such a vortex, leading to the hypothesis that each chemical element corresponds to a different kind of knot. The simple toroidal vortex, represented by the
circular “unknot” 01, was thought to represent hydrogen. Many elements had yet to be discovered, so the next knot, the trefoil knot 31, was thought to represent carbon.
However, as more elements were discovered and the periodicity of their characteristics established in the periodic table of the elements, it became clear that this could not be explained by any
rational classification of knots. This, together with the discovery of subatomic particles such as the electron, led to the theory being abandoned. (Wikipedia)
Since we are discussing about prime distribution then this 18's structure will also cover the further scheme that is inherited from the above 37 files.
This web enabled demonstration shows a polar plot of the first 20 non-trivial Riemann zeta function zeros (including Gram points) along the critical line Zeta(1/2+it) for real values of t running
from 0 to 50. The consecutively labeled zeros have 50 red plot points between each, with zeros identified by concentric magenta rings scaled to show the relative distance between their values of t.
Gram’s law states that the curve usually crosses the real axis once between zeros. (TheoryOfEverything)
1 + 7 + 29 = 37 = 19 + 18
By our project, these 37 files are located within the wiki of main repository and organized by the 18's structure located per the 18 files of project gist.
Angular Momentum
You may learn that sets of algebraic objects has a multilinear relationship related to a vector space called tensor.
Tensors may map between different objects such as vectors, scalars, even other tensors contained in a group of partitions.
In mathematical physics, Clebsch–Gordan coefficients are the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis.
Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of
these irreducible representations are known abstractly. The name derives from the German mathematicians Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912), who encountered an equivalent problem
in invariant theory.
Generalization to SU(3) of Clebsch–Gordan coefficients is useful because of their utility in characterizing hadronic decays, where a flavor-SU(3) symmetry exists (the eightfold way) that connects the
three light quarks: up, down, and strange. (Wikipedia)
In linear algebra, there is vector is known as eigenvector, a nonzero vector that changes at most by a scalar factor when linear transformation is applied to it.
The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them (Wikipedia).
In later sections, we will discuss finding all the solutions to a polynomial function. We will also discuss solving multiple equations with multiple unknowns.
From what we learned above about segregating twin prime candidates, we can demonstrate that they compile additively in perfect progression, completing an infinite sequence of circles (multiples of 30
and 360)
Δ prime = 114th prime - 19 = (6 x 19)th prime - 19 = 619 - 19 = 600 = 3 x 200
Observing more detail of the discussed scheme of 168 we will get it also when we take the 19's and 17's cell of (31+37)+(35+65)=68+100=168.
Physical Movements
By our project the 18’s on the gist will cover five (5) unique functions that behave as one (1) central plus four (4) zones. This scheme will be implemented to all of the 168 repositories as
bilateral way (in-out) depend on their postion on the system. So along with the gist it self then there shall be 1 + 168 = 169 units of 1685 root functions.
5 + 2 x 5 x 168 = 5 + 1680 = 1685 root functions
By the spin above you can see that the 4 zones of these 19's to 17's are representing the rotation 1 to 5. Such of formation can be seen on Ulam Spiral as below.
The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner’s Mathematical Games column in
Scientific American a short time later.
By the MEC30 we will also discuss the relation of these 4 zones with high density of 40 primes where 60 number is folded.
Both Ulam and Gardner noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler’s
prime-generating polynomial x²-x+41, are believed to produce a high density of prime numbers. Nevertheless, the Ulam spiral is connected with major unsolved problems in number theory such as Landau’s
problems (Wikipedia).
So by the eight (8) pairs of prime it will always return to the beginning position within 60+40=100 nodes per layer.
The published diagram by Feynman helped scientists track particle movements in illustrations and visual equations rather than verbose explanations. What seemed almost improbable at the time is now
one of the greatest explanations of particle physics — the squiggly lines, diagrams, arrows, quarks, and cartoonish figures are now the established nomenclature and visual story that students,
scientists, and readers will see when they learn about this field of science. (medium.com)
8 pairs = 8 x 2 = 16
Transforming particles into anti-particles, and vice versa, requires only the complex conjugate i → −i in our formalism. (Standard Model from an algebra - pdf) | {"url":"https://www.eq19.com/grammar/multiplication/","timestamp":"2024-11-02T14:44:33Z","content_type":"application/xhtml+xml","content_length":"83372","record_id":"<urn:uuid:64794094-673a-4823-b9cb-f0d518acc813>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00160.warc.gz"} |
contracted_nodes(G, u, v, self_loops=True, copy=True)[source]#
Returns the graph that results from contracting u and v.
Node contraction identifies the two nodes as a single node incident to any edge that was incident to the original two nodes.
GNetworkX graph
The graph whose nodes will be contracted.
u, vnodes
Must be nodes in G.
If this is True, any edges joining u and v in G become self-loops on the new node in the returned graph.
If this is True (default True), make a copy of G and return that instead of directly changing G.
Networkx graph
If Copy is True, A new graph object of the same type as G (leaving G unmodified) with u and v identified in a single node. The right node v will be merged into the node u, so only u will
appear in the returned graph. If copy is False, Modifies G with u and v identified in a single node. The right node v will be merged into the node u, so only u will appear in the returned
For multigraphs, the edge keys for the realigned edges may not be the same as the edge keys for the old edges. This is natural because edge keys are unique only within each pair of nodes.
For non-multigraphs where u and v are adjacent to a third node w, the edge (v, w) will be contracted into the edge (u, w) with its attributes stored into a “contraction” attribute.
This function is also available as identified_nodes.
Contracting two nonadjacent nodes of the cycle graph on four nodes C_4 yields the path graph (ignoring parallel edges):
>>> G = nx.cycle_graph(4)
>>> M = nx.contracted_nodes(G, 1, 3)
>>> P3 = nx.path_graph(3)
>>> nx.is_isomorphic(M, P3)
>>> G = nx.MultiGraph(P3)
>>> M = nx.contracted_nodes(G, 0, 2)
>>> M.edges
MultiEdgeView([(0, 1, 0), (0, 1, 1)])
>>> G = nx.Graph([(1, 2), (2, 2)])
>>> H = nx.contracted_nodes(G, 1, 2, self_loops=False)
>>> list(H.nodes())
>>> list(H.edges())
[(1, 1)]
In a MultiDiGraph with a self loop, the in and out edges will be treated separately as edges, so while contracting a node which has a self loop the contraction will add multiple edges:
>>> G = nx.MultiDiGraph([(1, 2), (2, 2)])
>>> H = nx.contracted_nodes(G, 1, 2)
>>> list(H.edges()) # edge 1->2, 2->2, 2<-2 from the original Graph G
[(1, 1), (1, 1), (1, 1)]
>>> H = nx.contracted_nodes(G, 1, 2, self_loops=False)
>>> list(H.edges()) # edge 2->2, 2<-2 from the original Graph G
[(1, 1), (1, 1)] | {"url":"https://networkx.org/documentation/latest/reference/algorithms/generated/networkx.algorithms.minors.contracted_nodes.html","timestamp":"2024-11-06T04:54:18Z","content_type":"text/html","content_length":"41798","record_id":"<urn:uuid:46f0f81c-362a-4a31-b5f6-287a78d699f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00299.warc.gz"} |
Graphing Polynomials Worksheet Linear Quadratic Cubic Quartic Quintic - Graphworksheets.com
Graphing Polynomials Worksheet Linear Quadratic Cubic Quartic Quintic
Graphing Polynomials Worksheet Linear Quadratic Cubic Quartic Quintic – Line Graph Worksheets can help you develop your understanding of how a line graph works. There are different types of line
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A line graph is a useful tool for visualizing data. It can show trends and change over time. For example, it can show the rate at which bacteria grow, or the changes in temperature and pH levels.
These trends and patterns can help you make predictions about the future. Several different types of questions can be included in a line graph worksheet.
A line graph usually contains two axes: the X-axis, which represents horizontal data and the Y-axis, which shows vertical data. The X-axis is time and the Y axis, which is the vertical left-hand
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Line graph worksheets are a valuable tool for teaching statistical concepts. These worksheets provide students with ample practice, which allows them to draw and interpret line graphs. Besides
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Excel’s line and bar graph options make graphing data easy. Bar graphs are best for showing data points, trends, and proportions. Line graphs are best for showing data points over long periods of
time, but they can also be misleading. Incorrectly plotting data can lead to exaggeration or hiding of certain results.
Create a scatter plot
Line graph worksheets can be used to create scatter plots from data sets. These graphs have columns that contain independent and dependent variables. You can change the line color and size, and
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A scatter plot is a chart that displays two numeric values and shows the correlation between them. This type of graph usually has two columns. The dependent variable is shown on the Y-axis and the
independent variable on the X. These two values can be stacked together to create a timeline.
Scatter plots are used for predictive modeling. They can also be used to identify outliers in your data. If you’re studying advanced math and science, knowing how to interpret scatter plots will be
For a line graph, write a title
If you are looking to visualize changes over time, line graphs are the way to go. Line graphs are particularly useful for data with peaks or valleys, and they can be collected in a relatively short
time. A line graph’s title should be descriptive. Although you can use many words to describe the graph, it should be clear and concise.
In addition to graphing data over time, line graphs can also be used to compare two sets of information. For example, a graph might compare the cost of milk in 2005 with the cost in 2005. Then, the
student should identify the first point on the graph and connect it to the other points using lines.
The title on the graph can be placed above or below the graphical image. The title will not resize if placed below the graphical image.
Gallery of Graphing Polynomials Worksheet Linear Quadratic Cubic Quartic Quintic
Howto How To Factor Cubic Polynomials With 4 Terms
Polynomial Chart Polynomials Polynomial Graph Quadratics
Leave a Comment | {"url":"https://www.graphworksheets.com/graphing-polynomials-worksheet-linear-quadratic-cubic-quartic-quintic/","timestamp":"2024-11-02T20:49:48Z","content_type":"text/html","content_length":"61160","record_id":"<urn:uuid:afcd8439-15e7-4e8a-aa9f-f798fece37fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00300.warc.gz"} |
Plotting Expression of Differential Transcripts plotDEXseq
Last seen 2.2 years ago
I was wondering which counts are appropriate to use for plotting the expression of the differential transcripts? plotDEXSeq is giving me this error, even when using the pre-constructed matrix of
counts from the workflow "Swimming downstream: statistical analysis of differential transcript usage following Salmon quantification":
plotDEXSeq( dxr2, "ENSG00000000457.13", legend=FALSE, cex.axis=1.2, cex=1.3,
lwd=2 )
#legend is false because we have not imported everything from the txdf.
Error in plot.window(xlim = c(0, 1), ylim = c(0, max(matr))) :
need finite 'ylim' values
In addition: Warning messages:
1: In max(coeff, na.rm = TRUE) :
no non-missing arguments to max; returning -Inf
2: In max(matr) : no non-missing arguments to max; returning -Inf
I am interested in outputting all normalized transcripts that I found to be to differentially expressed.
Further, I have looked at the function plotDEXseq and it uses featureCounts, which is just like requesting in Deseq2, counts(object, normalized = TRUE) of the specific groupID. I can't help but
wondering if these counts are OK to graphically represent a *transcript* that was found to be differential expressed following DEXseq and StageR?
Thank You
Entering edit mode
Hi Mike,
I agree with you regarding the plotting function of DRIMSeq, especially the plotProportions function; however, when I try to run dmPercision on my entire dmDSdata and not a subset I receive the
error below:
An object of class dmDSdata
with 11120 genes and 32 samples
* data accessors: counts(), samples()
d <- DRIMSeq::dmPrecision(d, design=design_full)
! Using a subset of 0.1 genes to estimate common precision !
Error in optimHess(par = par, fn = dm_lik_regG, gr = dm_score_regG, x = x, :
non-finite value supplied by optim
In addition: There were 50 or more warnings (use warnings() to see the first 50)
Entering edit mode
My recommendation for now is to make simple plots on your own, and not to run DRIMSeq all over again if you've used DEXSeq.
Entering edit mode
OK. So that takes us back to my initial question: Which counts are appropriate to use for plotting the expression of the differential transcripts? Would this be correct to use, following DE of a
transcript(s) after running DEXSeq and StageR
count <- featureCounts(dxd, normalized = TRUE)[idx, ]
Entering edit mode
I’d recommend using scaledTPM from tximport for the reasons described in the workflow. | {"url":"https://support.bioconductor.org/p/114028/","timestamp":"2024-11-13T05:36:56Z","content_type":"text/html","content_length":"26456","record_id":"<urn:uuid:7451ea6f-b550-4960-a133-f7901f586611>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00583.warc.gz"} |
Distributive property of arithmetic
distributive property of arithmetic Related topics: difference of square
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Author Message
wevmedes Posted: Monday 06th of Oct 10:19
It’s really difficult for me to figure this out alone so I think I need someone to give an advice. I need help concerning distributive property of arithmetic. It’s giving me sleepless
nights every time I try to understand it because I just can’t seem to discover how to do it. I read some books about it but it’s really confusing . Can I ask help from someone of you
guys here? I require someone who can explain how to answer some questions concerning distributive property of arithmetic.
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AllejHat Posted: Tuesday 07th of Oct 11:24
That’s exactly the kind of problem I had faced . Can you enlighten a bit more on what your problems are with distributive property of arithmetic? Yeah . Getting an inexpensive coach
suitable to your requirements is somewhat easier said than done these days . But even I went on to do exactly all the things that you are doing now. But then, my hunt was over when I
found out that there are a number of programs in algebra. They are affordable . I was actually excited with it. May be this is just what you need. What do you think about this?
From: Odense,
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malhus_pitruh Posted: Tuesday 07th of Oct 21:42
I agree. Stress will lead you to doom . Algebrator is a very useful tool. You don’t need to be a computer pro in order to use it. Its simple to use, and it works great.
From: Girona,
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Dnexiam Posted: Thursday 09th of Oct 17:42
algebra formulas, side-angle-side similarity and inequalities were a nightmare for me until I found Algebrator, which is truly the best math program that I have come across. I have
used it frequently through many algebra classes – Remedial Algebra, Algebra 1 and Pre Algebra. Just typing in the math problem and clicking on Solve, Algebrator generates step-by-step
solution to the problem, and my algebra homework would be ready. I really recommend the program.
From: City 17
Back to top | {"url":"https://www.softmath.com/algebra-software-2/distributive-property-of.html","timestamp":"2024-11-11T14:13:27Z","content_type":"text/html","content_length":"39355","record_id":"<urn:uuid:927d7cb3-e354-4e86-a965-6b7184ccf635>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00548.warc.gz"} |
| Tool Preview
Mothur Overview
Mothur is a comprehensive suite of tools for microbial ecology community. It is initiated by Dr. Patrick Schloss and his software development team in the Department of Microbiology and Immunology at
The University of Michigan. For more information, see Mothur-Wiki.
Command Documentation
The classify.otu command assigns sequences to chosen taxonomy outline.
The basis parameter allows you indicate what you want the summary file to represent, options are otu and sequence. Default is otu. For example consider the following basis=sequence could give
Clostridiales 3 105 16 43 46, where 105 is the total number of sequences whose otu classified to Clostridiales. 16 is the number of sequences in the otus from groupA, 43 is the number of sequences in
the otus from groupB, and 46 is the number of sequences in the otus from groupC. Now for basis=otu could give Clostridiales 3 7 6 1 2, where 7 is the number of otus that classified to Clostridiales.
6 is the number of otus containing sequences from groupA, 1 is the number of otus containing sequences from groupB, and 2 is the number of otus containing sequences from groupC.
v1.21.0: Updated to use Mothur 1.33. Added count parameter (1.28.0) and persample parameter (1.29.0) | {"url":"https://toolshed.g2.bx.psu.edu/repository/display_tool?repository_id=cde902adbb8f6da5&tool_config=%2Fsrv%2Ftoolshed%2Fmain%2Fvar%2Frepos%2F003%2Frepo_3215%2Fclassify.otu.xml&changeset_revision=ff1bc0299372&render_repository_actions_for=tool_shed","timestamp":"2024-11-12T02:22:22Z","content_type":"text/html","content_length":"9573","record_id":"<urn:uuid:9435e4ff-d4f3-46e0-997a-bc06179ee1c0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00060.warc.gz"} |
Neural Networks with Use Cases
Simple Definition Of A Neural Network
Modeled in accordance with the human brain, a Neural Network was built to mimic the functionality of a human brain. The human brain is a neural network made up of multiple neurons, similarly, an
Artificial Neural Network (ANN) is made up of multiple perceptrons (explained later).
Neural Network Architecture
The Neural Network is constructed from 3 type of layers:
1. Input layer — initial data for the neural network.
2. Hidden layers — intermediate layer between input and output layer and place where all the computation is done.
3. Output layer — produce the result for given inputs.
There are 3 yellow circles on the image above. They represent the input layer and usually are noted as vector X. There are 4 blue and 4 green circles that represent the hidden layers. These circles
represent the “activation” nodes and usually are noted as W or θ. The red circle is the output layer or the predicted value (or values in case of multiple output classes/types).
Each node is connected with each node from the next layer and each connection (black arrow) has particular weight. Weight can be seen as impact that node has on the node from the next layer.
What Is Deep Learning?
Deep Learning is an advanced field of Machine Learning that uses the concepts of Neural Networks to solve highly-computational use cases that involve the analysis of multi-dimensional data. It
automates the process of feature extraction, making sure that very minimal human intervention is needed.
Deep Learning Use Case/ Applications
Did you know that PayPal processes over $235 billion in payments from four billion transactions by its more than 170 million customers? It uses this vast amount of data to identify possible
fraudulent activities among other reasons.
With the help of Deep Learning algorithms, PayPal mined data from their customer’s purchasing history in addition to reviewing patterns of likely fraud stored in its databases to predict whether a
particular transaction is fraudulent or not.
The company has been relying on Deep Learning & Machine Learning technology for around 10 years. Initially, the fraud monitoring team used simple, linear models. But over the years the company
switched to a more advanced Machine Learning technology called, Deep Learning.
Fraud risk manager and Data Scientist at PayPal, Ke Wang, quoted:
Ke Wang,Fraud Risk Manager,PayPal
“What we enjoy from more modern, advanced machine learning is its ability to consume a lot more data, handle layers and layers of abstraction and be able to ‘see’ things that a simpler technology
would not be able to see, even human beings might not be able to see.”
A simple linear model is capable of consuming around 20 variables. However, with Deep Learning technology one can run thousands of data points. Therefore, by implementing Deep Learning technology,
PayPal can finally analyze millions of transactions to identify any fraudulent activity.
How Does A Neural Network Work?
To understand neural networks, we need to break it down and understand the most basic unit of a Neural Network, i.e. a Perceptron.
What Is A Perceptron?
A Perceptron is a single layer neural network that is used to classify linear data. It has 4 important components:
1. Inputs
2. Weights and Bias
3. Summation Function
4. Activation or transformation Function
The basic logic behind a Perceptron is as follows:
The inputs (x) received from the input layer are multiplied with their assigned weights w. The multiplied values are then added to form the Weighted Sum. The weighted sum of the inputs and their
respective weights are then applied to a relevant Activation Function. The activation function maps the input to the respective output.
Types of Artificial Neural Networks
There are two important types of Artificial Neural Networks –
• FeedForward Neural Network
• FeedBack Neural Network
FeedForward Artificial Neural Networks
In the feedforward ANNs, the flow of information takes place only in one direction. That is, the flow of information is from the input layer to the hidden layer and finally to the output. There are
no feedback loops present in this neural network. These type of neural networks are mostly used in supervised learning for instances such as classification, image recognition etc. We use them in
cases where the data is not sequential in nature.
Feedback Artificial Neural Networks
In the feedback ANNs, the feedback loops are a part of it. Such type of neural networks are mainly for memory retention such as in the case of recurrent neural networks. These types of networks are
most suited for areas where the data is sequential or time-dependent.
Bayesian Networks
These type of neural networks have a probabilistic graphical model that makes use of Bayesian Inference for computing the probability. These type of Bayesian Networks are also known as Belief
Networks. In these Bayesian Networks, there are edges that connect the nodes representing the probabilistic dependencies present among these type of random variables. The direction of effect is such
that if one node is affecting the other then they fall in the same line of effect. Probability associated with each node quantifies the strength of the relationship. Based on the relationship, one is
able to infer from the random variables in the graph with the help of various factors.
The only constraint that these networks have to follow is it cannot return to the node through the directed arcs. Therefore, Bayesian Networks are referred to as Directed Acyclic Graphs (DAGs).
Neural Networks Explained With An Example
Consider a scenario where you are to build an Artificial Neural Network (ANN) that classifies images into two classes:
• Class A: Containing images of non-diseased leaves
• Class B: Containing images of diseased leaves
So how do you create a Neural network that classifies the leaves into diseased and non-diseased crops?
The process always begins with processing and transforming the input in such a way that it can be easily processed. In our case, each leaf image will be broken down into pixels depending on the
dimension of the image.
For example, if the image is composed of 30 by 30 pixels, then the total number of pixels will be 900. These pixels are represented as matrices, which are then fed into the input layer of the Neural
Just like how our brains have neurons that help in building and connecting thoughts, an ANN has perceptrons that accept inputs and process them by passing them on from the input layer to the hidden
and finally the output layer.
As the input is passed from the input layer to the hidden layer, an initial random weight is assigned to each input. The inputs are then multiplied with their corresponding weights and their sum is
sent as input to the next hidden layer.
Here, a numerical value called bias is assigned to each perceptron, which is associated with the weightage of each input. Further, each perceptron is passed through activation or a transformation
function that determines whether a particular perceptron gets activated or not.
An activated perceptron is used to transmit data to the next layer. In this manner, the data is propagated (Forward propagation) through the neural network until the perceptrons reach the output
At the output layer, a probability is derived which decides whether the data belongs to class A or class B.
Artificial Neural Networks Applications
Following are the important Artificial Neural Networks applications –
Handwritten Character Recognition
ANNs are used for handwritten character recognition. Neural Networks are trained to recognize the handwritten characters which can be in the form of letters or digits.
Speech Recognition
ANNs play an important role in speech recognition. The earlier models of Speech Recognition were based on statistical models like Hidden Markov Models. With the advent of deep learning, various types
of neural networks are the absolute choice for obtaining an accurate classification.
Signature Classification
For recognizing signatures and categorizing them to the person’s class, we use artificial neural networks for building these systems for authentication. Furthermore, neural networks can also classify
if the signature is fake or not.
Facial Recognition
In order to recognize the faces based on the identity of the person, we make use of neural networks. They are most commonly used in areas where the users require security access. Convolutional Neural
Networks are the most popular type of ANN used in this field.
So, you saw the use of artificial neural networks through different applications. We studied how neural networks are able to predict accurately using the process of backpropagation. We also went
through the Bayesian Networks and finally, we overviewed the various applications of ANNs. | {"url":"https://rutujakonde210.medium.com/neural-networks-with-use-cases-bf2bfa15d43f?source=user_profile_page---------8-------------cffc5ff6c780---------------","timestamp":"2024-11-05T12:03:45Z","content_type":"text/html","content_length":"163171","record_id":"<urn:uuid:15ceef5d-80bb-43eb-9c08-828a36ead53a>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00387.warc.gz"} |
Generating Synthetic Data with Transformers: A Solution for Enterprise Data Challenges | NVIDIA Technical Blog
Big data, new algorithms, and fast computation are three main factors that make the modern AI revolution possible. However, data poses many challenges for enterprises: difficulty in data labeling,
ineffective data governance, limited data availability, data privacy, and so on.
Synthetically generated data is a potential solution to address these challenges because it generates data points by sampling from the model. Continuous sampling can generate an infinite number of
data points including labels. This allows for data to be shared across teams or externally.
Generating synthetic data also provides a degree of data privacy without compromising quality or realism. Successful synthetic data generation involves capturing the distribution while maintaining
privacy and conditionally generating new data, which can then be used to make more robust models or used for time-series forecasting.
In this post, we explain how synthetic data can be artificially produced with transformer models, using NVIDIA NeMo as an example. We explain how synthetically generated data can be used as a valid
substitute for real-life data in machine learning algorithms to protect user privacy while making accurate predictions.
Transformers: the better synthetic data generator
Deep learning generative models are a natural fit to model complicated real-world data. Two popular generative models have achieved some success in the past: Variational Auto-Encoder (VAE) and
Generative Adversarial Network (GAN).
However, there are known issues with VAE and GAN models for synthetic data generation:
• The mode collapse problem in the GAN model causes the generated data to miss some modes in the training data distribution.
• The VAE model has difficulty generating sharp data points due to non-autoregressive loss.
Transformer models have recently achieved great success in the natural language processing (NLP) domain. The self-attention encoding and decoding architecture of the transformer model has proven to
be accurate in modeling data distribution and is scalable to larger datasets. For example, the NVIDIA Megatron-Turing NLG model obtains excellent results with 530B parameters.
OpenAI’s GPT3 uses the decoder part of the transformer model and has 175B parameters. GPT3 has been widely used across multiple industries and domains, from productivity and education to creativity
and games.
The GPT model turns out to be a superior generative model. As you may know, any joint probability distribution can be factored into the product of a series of conditional probability distributions
according to the probability chain rule. The GPT autoregressive loss directly models the data joint probability distribution shown in Figure 1.
Figure 1. GPT model training
In Figure 1, the GPT model training uses autoregressive loss. It has a one-to-one mapping to the probability chain rule. GPT directly models the data joint probability distribution.
Because tabular data is composed of different types of data as rows or columns, GPT can understand the joint data distribution across multiple table rows and columns, and generate synthetic data as
if it were NLP-textual data. Our experiments show that indeed the GPT model generates higher-quality tabular synthetic data.
A higher-quality tabular data tokenizer
Despite its superiority, there are a number of challenges with using GPT to model tabular data: the data inputs to the GPT model are sequences of token IDs. For NLP datasets, you could use a
byte-pair encoding (BPE) tokenizer to convert the text data into sequences of token IDs.
It is natural to use the generic GPT BPE tokenizer for tabular datasets; however, there are a few problems with this approach.
First, when the GPT BPE tokenizer splits the tabular data into tokens, the number of tokens is usually not fixed for the same column at different rows, because the number is determined by the
occurrence frequencies of the individual subtokens. This means that the columnar information in the table is lost if you use an ordinary NLP tokenizer.
Another problem with the NLP tokenizer is that a long string in a column would consist of a large number of tokens. This is wasteful considering that GPT has a limited capacity for modeling the
sequences of tokens. For example, the merchant name Mitsui Engineering & Shipbuilding Co needs 7 tokens to encode it ([44, 896, 9019, 14044, 1222, 16656, 16894, 1766]) with a BPE tokenizer.
As discussed in the TabFormer paper, a viable solution is to build a specialized tokenizer for the tabular data that considers the table’s structural information. The TabFormer tokenizer uses a
single token for each of the columns, which can cause either accuracy loss if the number of tokens is small for the column, or weak generalization if the number of tokens is too large.
We improve it by using multiple tokens to code the columns.
Figure 2. Convert float numbers into a sequence of token IDs
Figure 2 shows the steps of converting a float number into a sequence of token IDs. First, we reversibly convert the float number into a positive integer. Then, it is transformed into a number with
positional base B, where B is a hyperparameter. The larger the base B number is, the fewer tokens it needs to represent the number.
However, a larger base B sacrifices the generality for new numbers. In the last step, the digit numbers are mapped to unique token IDs. To convert the token IDs to a float number, run through these
steps in reverse order. The float number decoding accuracy is then determined by the number of tokens and the choice of positional base B.
Scaling model training with NeMo framework
NeMo is a framework for training conversational AI models. In the released code inside the NeMo repository, our tabular data tokenizer supports both integer and categorical data, handles NaN values,
and supports different scalar transformations to minimize the gaps between the numbers. For more information, see our source code implementation.
You can use the special tabular data tokenizer to train a tabular synthetic data generation GPT model of any size. Large models can be difficult to train due to memory constraints. The NeMo framework
is a toolkit for training large language models within NeMo and provides both tensor model parallel and pipeline model parallelism.
This enables the training of transformer models with billions of parameters. On top of the model parallelism, you can apply data parallelism during training to fully use all GPUs in the cluster.
According to OpenAI’s scaling law of natural language and theory of over-parameterization of deep learning models, it is recommended to train a large model to get reasonable validation loss given the
training data size.
Applying GPT models to real-world applications
In our recent GTC talk, we showed that a trained large GPT model produces high-quality synthetic data. If we continue sampling the trained tabular GPT model, it can produce an infinite number of data
points, which all follow the joint distribution as the original data. The generated synthetic data provides the same analytical insights as the original data without revealing the individual’s
private information. This makes safe data sharing possible.
Moreover, if you condition the generative model on past data to generate future synthetic data, the model is actually predicting the future. This is attractive to customers in the financial services
industry who are dealing with financial time series data. In collaboration with Cohen & Steers, we implemented a tabular GPT model to forecast economic and market indicators including inflation,
volatility, and equity markets with quality results.
Bloomberg presented at GTC 2022 how they applied our proposed synthetic data method to analyze the patterns of credit card transaction data while protecting user data privacy.
Apply your knowledge
In this post, we introduced the idea of using NeMo for synthetic tabular data generation and showed how it can be used to solve real-world problems. For more information, see The Data-centric AI
If you are interested in applying this technique to your own synthetic data generation, use this NeMo framework Synthetic Tabular Data Generation notebook tutorial. For hands-on training on applying
this method to generate synthetic data, reach out to us directly.
For more information, see the following GTC sessions: | {"url":"https://developer.nvidia.com/blog/generating-synthetic-data-with-transformers-a-solution-for-enterprise-data-challenges/","timestamp":"2024-11-15T04:06:37Z","content_type":"text/html","content_length":"215227","record_id":"<urn:uuid:e8df0e26-255c-448d-ac7c-d3301ebae285>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00336.warc.gz"} |
Algebra (NCTM)
Represent and analyze mathematical situations and structures using algebraic symbols.
Recognize and generate equivalent forms for simple algebraic expressions and solve linear equations
Grade 6 Curriculum Focal Points (NCTM)
Algebra: Writing, interpreting, and using mathematical expressions and equations
Students write mathematical expressions and equations that correspond to given situations, they evaluate expressions, and they use expressions and formulas to solve problems. They understand that
variables represent numbers whose exact values are not yet specified, and they use variables appropriately. Students understand that expressions in different forms can be equivalent, and they can
rewrite an expression to represent a quantity in a different way (e.g., to make it more compact or to feature different information). Students know that the solutions of an equation are the values of
the variables that make the equation true. They solve simple one-step equations by using number sense, properties of operations, and the idea of maintaining equality on both sides of an equation.
They construct and analyze tables (e.g., to show quantities that are in equivalent ratios), and they use equations to describe simple relationships (such as 3x = y) shown in a table. | {"url":"https://newpathworksheets.com/math/grade-6/algebraic-equations-1?dictionary=word+problems&did=304","timestamp":"2024-11-08T05:44:24Z","content_type":"text/html","content_length":"47386","record_id":"<urn:uuid:98ef0c05-0aaf-4c1b-b74b-aa2f830a0e5a>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00208.warc.gz"} |
| Analytic Trigonometry
Hyperbolic functions [Solved!]
Matt 25 Nov 2015, 09:42
My question
I have this problem to solve:
1 + (e2A-e-2A/2) + (e2A+e-2A/2)
1 - (e2A-e-2A/2) - (e2A+e-2A/2)
The (2A) is are the the power of (e)
Can you help me simplify this please?
Relevant page
What I've done so far
Expanded it out and tried to simplfy, but couldnt do it | {"url":"https://www.intmath.com/forum/analytic-trigonometry-25/hyperbolic-functions:17","timestamp":"2024-11-08T14:52:41Z","content_type":"text/html","content_length":"103562","record_id":"<urn:uuid:1a243290-7d31-4d09-a211-d3d7f191bd04>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00365.warc.gz"} |
PPT - Harvey Newman & Ren -Yuan Zhu California Institute of Technology March 27, 2013 PowerPoint Presentation - ID:628727
Télécharger la présentation
Harvey Newman & Ren -Yuan Zhu California Institute of Technology March 27, 2013
An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed
/ shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While
downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file
might be deleted by the publisher. | {"url":"https://fr.slideserve.com/elinor/mindrum-precision-inc","timestamp":"2024-11-13T03:23:25Z","content_type":"text/html","content_length":"84571","record_id":"<urn:uuid:1bb3b518-d65d-4ece-a900-c22b56218b2b>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00730.warc.gz"} |
Define Employee Category in Tally
In Tally, an employee category classifies the company employees, which is based on the projects or its locations.
Create Single Employee Category
Use the following step by step process to create a single employee category.
Gateway of Tally → Payroll Info → Employee Categories → Single Category → Create
Step 2: Choose the option Payroll Info under the Gateway of Tally.
Step 3: Choose employee categories under payroll info features, as shown below.
Step 4: Choose the “Create” option under a single category to create a single employee in Tally.
Step 5: Define the following details on employee category creation.
Name: Define the employee name category, which is to be created in Tally ERP 9.
Allocate revenue items: To assign the revenue related transaction values for the employee, choose the option as “Yes”.
Allocate non-revenue items: To assign the non-revenue related transaction values for employees, choose the option as “Yes”.
To save the details in Tally, choose A: Accept.
Create Multiple Employee Categories
Step 1: Use the following path to create multiple employee categories.
Gateway of Tally → Payroll Info → Employee Categories → Multiple Categories → Create
Step 2: Choose the “Create” option under multiple categories.
Step 3: Update the following details on multi employee’s creation screen, as shown below:
To save the details in Tally, choose A: Accept. | {"url":"https://ncert-books.com/define-employee-category-in-tally/","timestamp":"2024-11-07T04:31:00Z","content_type":"text/html","content_length":"121380","record_id":"<urn:uuid:343b0e30-59d2-4441-8cce-2022dc20d52e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00218.warc.gz"} |
Replacing strings using Regular Expression Regex - FcukTheCode
Replacing strings using Regular Expression Regex
Basics of Regular Expressions in Python, check the link below π π
Regular Expressions Regex basic in Python π π
Also with table of contents about symbols and their usage in Regex
Links for other methods like search, flags and pre-compiled patterns in regular expression scroll down
π π
Python makes regular expressions available through the re module.
Replacing in Regex Python
Replacements can be made on strings using re.sub .
Replacing strings
import re
print( re.sub(r"t[0-9][0-9]", "foo", "my name t13 is t44 what t99 ever t44") )
# Output: 'my name foo is foo what foo ever foo'
OUTPUT: my name foo is foo what foo ever foo
Using group references
Replacements with a small number of groups can be made as follows:
import re
print( re.sub(r"t([0-9])([0-9])", r"t\2\1", "t13 t19 t81 t25") )
# Output: 't31 t91 t18 t52'
OUTPUT: t31 t91 t18 t52
However, if you make a group ID like ’10’, this doesn’t work: \10 is read as ‘ID number 1 followed by 0’. So you have to be more specific and use the \g notation:
import re
print( re.sub(r"t([0-9])([0-9])", r"t\g<2>\g<1>", "t13 t19 t81 t25") )
# Output: 't31 t91 t18 t52'
OUTPUT: t31 t91 t18 t52
Using a replacement function
import re
items = ["zero", "one", "two"]
print( re.sub(r"a\[([0-3])\]", lambda match: items[int(match.group(1))], "Items: a[0], a[1], something, a[2]") )
# Output: 'Items: zero, one, something, two'
OUTPUT: Items: zero, one, something, two
Executed using python3 linux terminal
For the first part in using regular expressions and matching the string
you can visit this link β > π π
Matching the beginning of a string (Regex) Regular Expressions in python
The re.search() method takes a regular expression pattern and a string and searches for that pattern within the string. If the search is successful, search() returns a match object or None
For the second part in using regular expression and searching the string
visit β > Searching β Regular Expressions (Regex) in Python
Precompiled_pattern β > π π
Precompiled patterns β Regular Expression(Regex) in Python
Compiling a pattern allows it to be reused later on in a program.
However, note that Python caches recently-used expressions, so β programs that use only a few regular expressions at a time neednβ t worry about compiling regular expressionsβ .
Flags in Regex π π Flags in Regular Expressions ( Regex ) in Pythonβ ¦.FTC
For some special cases we need to change the behavior of the Regular Expression, this is done using flags. Flags can be set in two ways, through the flags keyword or directly in the expression.
Morae Q! | {"url":"https://www.fcukthecode.com/ftc-replacing-in-regular-expression-regex-in-python-fcukthecode-com/","timestamp":"2024-11-09T10:14:04Z","content_type":"text/html","content_length":"158484","record_id":"<urn:uuid:438ba268-4065-4ce0-a6b4-b6e9bfcdd056>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00827.warc.gz"} |
Distribution of Charges in a Conductor and Action at Points: Solved Example Problems
Distribution of charges in a conductor: Solved Example Problems
EXAMPLE 1.23
Two conducting spheres of radius r1 = 8 cm and r2 = 2 cm are separated by a distance much larger than 8 cm and are connected by a thin conducting wire as shown in the figure. A total charge of Q =
+100 nC is placed on one of the spheres. After a fraction of a second, the charge Q is redistributed and both the spheres attain electrostatic equilibrium.
(a) Calculate the charge and surface charge density on each sphere.
(b) Calculate the potential at the surface of each sphere.
(a) The electrostatic potential on the surface of the sphere A is
The electrostatic potential on the surface of the sphere A is
Since VA = VB. We have
But from the conservation of total charge, Q = q1 + q2, we get q1 = Q – q2. By substituting this in the above equation,
Note that the surface charge density is greater on the smaller sphere compared to the larger sphere (σ2 ≈ 4σ1) which confirms the result σ1 / σ1= r2 / r2.
The potential on both spheres is the same. So we can calculate the potential on any one of the spheres.
Van de Graaff Generator: Solved Example Problems
EXAMPLE 1.24
Dielectric strength of air is 3 × 106 V m-1. Suppose the radius of a hollow sphere in the Van de Graff generator is R = 0.5 m, calculate the maximum potential difference created by this Van de
Graaff generator.
The electric field on the surface of the sphere (by Gauss law) is given by
The potential on the surface of the hollow metallic sphere is given by
with Vmax = EmaxR
Here Emax = 3 ×106 V/m . So the maximum potential difference created is given by
Vmax = 3 × 106 × 0.5
= 1.5 × 106 V (or) 1.5 million volt
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Physics : Electrostatics : Distribution of Charges in a Conductor and Action at Points: Solved Example Problems | | {"url":"https://www.brainkart.com/article/Distribution-of-Charges-in-a-Conductor-and-Action-at-Points--Solved-Example-Problems_38400/","timestamp":"2024-11-10T16:23:50Z","content_type":"text/html","content_length":"45103","record_id":"<urn:uuid:581edde8-5271-47d9-bba6-8f20aba28645>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00609.warc.gz"} |
Transforming the Roots of a Quadratic Equation
If we have a known quadratic equation, we can solve it to find the roots
The roots of a quadratic are in general, complex numbers, so this equation is written in terms of
If we transform the roots to give new numbers
If fact we often do not need to know
Example: The roots of
From (1)
Substituting the values of | {"url":"https://astarmathsandphysics.com/a-level-maths-notes/fp1/3444-transforming-the-roots-of-a-quadratic-equation.html?tmpl=component&print=1","timestamp":"2024-11-10T07:44:21Z","content_type":"text/html","content_length":"12825","record_id":"<urn:uuid:ea0a9370-b95a-44c5-b06e-afdc8cf8f9e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00673.warc.gz"} |
The Hulk in Free Fall
In contrast to Simon Stagg counting up to the well-known and often published escape velocity, here we have a genuine motion situation that would be familiar to any beginning student. One slight
issue: the answer is wrong (both from the math as solved and from the reality of the situation we’re facing). That does not detract from the fact that this one panel is as revolutionary in its own
way as the introduction of Wolverine or the birth of Phoenix.
Remember “Exact flying time withheld at request of Reed Richards!” in Fantastic Four #12? We’re going to show what the power of physics and mathematics can do with very little information.
Bruce Banner is falling eight miles towards earth through the atmosphere.
If there were no atmosphere, how long would it take and what would be his velocity at the time of impact?
Let’s look at these numbers.
First to figure out how fast Hulk will be going at impact we need to solve for the total time he’s in the air. We also need to keep our units straight. This is one of the best arguments for the
metric system you will ever see.
Distance = ½ acceleration x time^2
8 miles x 5280 feet / mile = ½ x 32 feet/sec^2 x time^2
51.4 seconds = time to Hulk’s impact.
So we should have less than a minute to Hulk’s hitting ground. If Hulk were falling through the vacuum of space to a planet with the same gravitational constant as earth’s it would be spot-on
Now we can use the relationship that velocity is equal to constant acceleration multiplied by time to solve for his speed at impact time.
Velocity = acceleration x time to get our impact velocity in miles per hour
Velocity = 32 feet/second^2 x 51.4 seconds x 1 mile/5280 feet x 3600 seconds/hour
Velocity = 1100 miles per hour
The text is off by a factor of 10.
The lazy …errrrr… experienced physicist would look at energy and realize that the total potential energy of an object 8 miles above the surface would be converted entirely into kinetic energy at the
surface and that immediately gives the answer. Potential energy near enough to the earth is equal to mass times height times the acceleration of gravity. Kinetic energy is one-half time mass times
the velocity squared. Giving us this relationship:
Mass cancels out on both sides of this equation^[1] (as Galileo observed and Newton proved it should) to get velocity
= 1100 miles per hour
Two different methods of calculating the final velocity give us the same result. Huzzah!
Wait a minute you say! A heavier object does NOT fall faster than a light object? Yes. Yes I do say that. But don’t trust me or anyone else on that. Get experimental data. Consider that Women’s
Single Luge and Men’s Double luge use exactly the same track. The course record speed for one woman at Lake Placid, NY is 43.985 seconds. Two men made the same distance in 43.641 seconds[2].
The Hulk’s mass dropping out of this equation also means that the biological gamma transformation he undergoes makes no difference in the calculation^[3]. Bruce Banner goes from a more-or-less 80
kilogram man to a 600-plus kilogram rampaging behemoth and we do not take into account where that additional mass comes from, while being concerned about the terminal velocity of his fall is the
dynamic of comic book mundane science vs. super-science in microcosm.
Now let’s turn our attention to the real-world consideration of air resistance, which any skydiver or paratrooper will tell you is significant. In general, it means that an average-sized human will
reach a terminal velocity and stop accelerating. Experimental evidence shows that terminal velocity is about 200 km/hour.
Using the relationship velocity=acceleration x time as we did earlier (and keeping our units consistent), Hulk will reach terminal velocity in about 5.7 seconds, and travel 160 meters in the
process, or 1.2% of his total fall of 12.9 kilometers. We’re making some simplifying assumptions and approximations here. Were Hulk a returning spacecraft or a parachuting airman we’d be working
some much more intricate math.
But this leads us to a total time to landing of about 232 seconds or roughly 4 minutes, using
Colonel Joseph Kittinger (who for years had the high-altitude free-fall record) fell from 31,300 meters (almost three times Hulk’s fall) to 5,500 meters before opening his parachute. It took
Kittinger roughly four minutes to go this distance (higher up he had much less air resistance) and he hit a maximum recorded speed of 988 km/hour.
Our calculated numbers for the Hulk are in rough agreement with this historic experimental result.
It is WAY more interesting to look at total energy of the Hulk, which would be mgh (mass x local gravity constant x height), assuming 600 kg for the Hulk and 9.8 m/sec^2 for gravity and a height of 8
miles = 12.9 km.
In the lack of air resistance, he would land with full potential energy of (after converting units appropriately) 7.5 x 10^6 Joules. 1 Megajoule (1 million Joules or 10^6 J) is roughly equivalent to
a stick of TNT — meaning the Hulk’s landing would be equivalent to about 7 sticks of TNT.
But air resistance means everything slows down. For a human this reduces the speed in the atmosphere to a final terminal velocity of roughly 200 kilometers per hour or 120 miles per hour. You can
increase this speed by pointing your head down like a missile and bringing your arms in.
But figuring terminal velocity of about 200 kilometers per hour, Hulk would land with an energy of
900,000 Joules = 900,000
With air resistance the math tells us Hulk lands with an impact energy equivalent to about a stick of dynamite. This sounds about correct.
The thing to note here is that the Hulk’s potential energy at that height does not vanish – the deficit of about six sticks of dynamite goes into moving the atmosphere.
Given what we said earlier, could that energy have gone into making up the Hulk’s bulk as he transforms?
Let’s find out. The relationship of energy and mass is well-established by the Twentieth Century’s most photogenic hair-challenged theoretician:
Note that in algebra class your teacher will tell you this equation should be written with the constant as a coefficient. But some things are worth violating convention for.
So the amount of mass that 7.5 – 0.9 = 6.6 Megajoules can produce given that the speed of light is 3 x 10^8 meters / second is 7.3 x 10^-9 kilograms. Not anywhere near enough to make up the body mass
differential. The mass of a speck of dust^[4] is about 7.5 x 10^-10 kg, so that would be about enough energy to add ten dust specks to Bruce Banner’s mass.
Since mass and energy are directly related, let’s look at how much raw energy it would take to create the Hulk’s mass. The amount of mass converted into energy in a 20 kiloton explosion (like say, in
the opening act of the Atomic Age), roughly equals 1 gram, which according to the US Bureau of Engraving and Printing is also the mass of a dollar bill. Ergo, the additional 500 Kg of mass Bruce
Banner develops in becoming the Hulk would be equivalent to the energy of 500,000 Hiroshima-sized explosions. Those cool food replicators on Star Trek that make your cup of tomato soup out of pure
energy? They’re marshalling the equivalent of a hydrogen bomb to put that together. Sometimes old-fashioned wet chemistry is a less expensive solution.
Jim Shooter, Editor-in-Chief at Marvel from 1978 to 1987, tells of being at a convention where Chris Claremont presented this adventure. Larry Niven, Science Fiction author and trained
mathematician, was in the audience and praised the work but noted the need to take terminal velocity into account. Larry Niven confirms the event happened.
Jim Shooter’s scientific bona fides were established early on when he won a science fair contest with a project on photosynthesis including a ball and stick model of the chlorophyll molecule with
marshmallows, dye, and toothpicks. Chlorophyll is not a simple molecule to model. In fact, its full stereochemistry had not been determined until a few years later in 1967. This led him into an
internship in a laboratory at the University of Pittsburgh.
[1] And unlike the Anti-Life Equation – this IS an equation.
[2] From the Team USA Luge site. They have a list of all luge tracks (there are surprisingly few in the world!). https://www.teamusa.org/USA-Luge/Luge-Tracks/Lake-Placid-NY
[3] Transforming into the Hulk the cross-sectional area and aerodynamics might change – but that is a more specialized topic in aerodynamics and fluid dynamics than the relatively simple kinematics
problem we’re presented with.
[4] https://www.mvorganizing.org/how-much-do-particles-weigh/ | {"url":"https://www.quidditch.com/the-hulk-in-free-fall/","timestamp":"2024-11-02T01:20:56Z","content_type":"text/html","content_length":"48259","record_id":"<urn:uuid:66a17487-7e9e-495d-a85f-56d8cd9b9d7e>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00248.warc.gz"} |
ules of thumb for knowledge acquisition from redundant data
The 80-20 rule or Pareto principle states that, often in life, 20% of effort can roughly achieve 80% of the desired effects. An interesting question is as to weather this rule also holds in the
context of information acquisition from redundant data that follows a power law frequency distribution. Can we learn 80% of URLs on the Web by parsing only 20% of the web pages? Can we learn 80% of
the used vocabulary by looking at only 20% of the tags? Can we learn 80% of the news by reading 20% of the newspapers? More generally, can we learn 80% of all available information in a corpus by
randomly sampling 20% of data (without replacement)?
Information Acquisition from redundant Data
We develop an abstract model of information acquisition from redundant data: We assume a random sampling process without replacement from an infinitely large data set which contains information with
bias. We are then interested in the fraction of total information we can expect to learn as function of (i) the sampled fraction (our "recall") and (ii) the bias of information (its "redundancy
We develop two rules of thumb with varying robustness. We first show that, when information bias follows a Zipf distribution, the 80-20 rule or Pareto principle does surprisingly not hold, and we can
rather expect to learn less than 40% of the information when randomly sampling 20% of the overall data. We then analytically prove that for large data sets, randomized sampling from power-law
distributions leads to "truncated distributions" with the same power-law exponent. This second rule is very robust and also holds for distributions that deviate substantially from a strict power law.
We further give one particular family of power-law functions that remain completely invariant under sampling.
• Rules of Thumb for Information Acquisition from Large and Redundant Data
Wolfgang Gatterbauer
ECIR 2011, pp. 479-490.
[Paper], [Presentation], [Presentation], [bib]
Full 40 page version with all proofs (arXiv:1012.3502 [cs.IR]): [Full version], [bib], (Version Dec 2010)
• Estimating Required Recall for Successful Knowledge Acquisition from the Web
Wolfgang Gatterbauer.
WWW 2006, pp. 969-970 (Poster Paper).
[Paper], [Poster], [Poster1+2], [bib]
• Wolfgang Gatterbauer (Associate Professor @ NEU)
This research is motivated by the VENTEX (Visualized Element Nodes Table EXtraction) project and was supported in part by a DOC scholarship from the Austrian Academy of Sciences, by the FIT-IT
program of the Austrian Federal Ministry for Transport, Innovation and Technology, and NSF grant IIS-0915054 (the BeliefDB project). Any opinions, findings, and conclusions or recommendations
expressed in this project are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or other agencies. | {"url":"http://uniquerecall.com/","timestamp":"2024-11-13T03:04:21Z","content_type":"text/html","content_length":"8643","record_id":"<urn:uuid:3ce6ab4e-e38b-424c-89fd-a3a39b275298>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00117.warc.gz"} |
Do You Know About the GIJSWIJT Sequence? It's Very Interesting » Mathematics Vedic Math School
Do You Know About the GIJSWIJT Sequence? It’s Very Interesting
Mathematics / By Prince Jha / 3 minutes of reading
The Gijswijt Sequence is named after DION GIJSWIJT by Neil Sloane. The Sequence is similar to the Kolakoski Sequence, but it counts the most extended blocks in terms of length.
Gijswijt Sequence Introduction:
Gijswijt’s Sequence can be defined as:
It is a self-describing sequence where the value of each term is equal to the maximum number of repeated blocks of numbers in the Sequence preceding the number.
DION GIJSWIJT by Neil Sloane
First Few Values of the Gijswijt Sequence
The first few values of GIJSWIJT’S Sequence are: –
1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 1, 1, 2, 2, 2, 3, 2, 1, …
Let us know the explanation of these terms: –
1. The first term of the Sequence is 1 by definition.
2. 1 in the second term shows the length 1 of the block of 1s.
3. 2 in the third term shows the length of 2 of the block of 1s in the first and second terms.
4. Now, you can see that the Sequence is decreasing at this point for the first time.
5. 1 in the fourth term represents the length 1 of the block of 2s in the 3rd term and the length 1 of the block “1, 2”.
6. There is no block of any repeated sequence immediately preceding the fourth term that is longer than length 1.
7. 1 in the fifth term shows that the length 1 of the “repeating” blocks “1” and “2, 1” and “1, 2, 1” and “1, 1, 2, 1” immediately precede the fifth term. None of these blocks are repeated more than
once, so the fifth term is 1.
8. And so on…………………….
Properties of Gijswijt’s Sequence
1. The slow rate of growth
2. Recursive structure
3. Similarity to Kolakoski Sequence
Note: – It has been proved that every natural number occurs in this Sequence at least once, but the Sequence has prolonged growth. The 220th term is 4, whereas the term 5 occurs in the Sequence near
the 10^10^23th term.
Do You Know
Practice Questions:
Practice questions related to the Gijswijt’s Sequence: –
1. A positive integer n is given to you. You have to find the first n terms of the Gijswijt Sequence.
Case 1 = If n is 3, which means
Input = 3
The output or the first n terms of the Sequence will be 1,1,2.
Case 2 = If n is 7, which means
Input = 7
The output will be, or the first n terms of the Sequence will be 1,1,2,1,1,2,2,
Practice: –
1. A positive integer n = 6 is given to you. What will be the first n terms of the GIJSWIJT’S Sequence?
The exploration of the Gijswijt sequence unveils a distinctive mathematical curiosity, showcasing the intriguing interplay between combinatorics and number theory. The Gijswijt sequence, with its
unique rules governing the arrangement of numbers, adds another layer of complexity to the realm of sequences. This sequence not only captures the imagination of mathematicians but also serves as a
testament to the endless possibilities within mathematical patterns.
It’s worth noting that, much like the Gijswijt sequence, other popular sequences have gained fame in mathematical circles. The well-known “Look-and-Say” sequence, with its self-referential structure,
and the minimal-redundancy Golomb sequence are among those captivating patterns.
These sequences, along with the Gijswijt sequence, contribute to the rich tapestry of mathematical exploration, offering challenges and insights that continue to captivate both seasoned
mathematicians and enthusiasts alike. As we navigate the intricate world of sequences, these patterns not only provide intellectual stimulation but also pave the way for further discoveries and
applications across various domains.
Leave a Comment Cancel Reply | {"url":"https://vedicmathschool.org/gijswijt-sequence/","timestamp":"2024-11-09T08:01:38Z","content_type":"text/html","content_length":"220556","record_id":"<urn:uuid:f8b5c042-249d-47fc-8f95-a87e4aa5ed84>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00802.warc.gz"} |
Percent Error
Accuracy and Precision
In lab, the accuracy of a given measurement will be limited by ___________________
________________________and __________________________________________
Practice: The density of a given substance is 2.1 g/ml. Three students measured the
density of the object in lab and there results are shown in the table below. Discuss
each student’s data with respect to accuracy and precision.
Student A
Student B
Student C
Student A_______________________________________________________
Student B________________________________________________________
Student C_________________________________________________________
Accuracy, Precision, and Percent Error
Percent Error
Percent error =
Percent error will always be a ______________number.
1. The boiling point of water is 100oC. During an experiment, water came to a boil at
97oC according to the thermometer that was being used. What is the percent
error of the thermometer?
2. An experiment was performed to determine the density of water. The results of
the experiment showed that water had a density of 1.15 g/mL. What was the
percent error in this experiment?
3. An experiment was conducted to find the mass of one mole of carbon atoms. The
results of the experiment showed that a mole of carbon atoms had a mass of
15.78 g. The accepted value of a mole of carbon atoms is 16.00 grams. What is
the percent error in this experiment?
4. An experiment performed to determine the density of lead yields a value of
10.95 g/cm3. The accepted value for the density of lead is 11.342 g/cm 3. Find the
percent error.
5. Find the percent error in a measurement of the boiling point of bromine if the
laboratory figure is 40.6oC and the accepted value is 59.35oC.
1. 3%
2. 15%
3. 1.375%
5. 31.6% | {"url":"https://studylib.net/doc/9189329/percent-error","timestamp":"2024-11-05T03:33:44Z","content_type":"text/html","content_length":"59405","record_id":"<urn:uuid:0e83c0e4-7a77-4b4d-aca1-66d806b94f5a>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00355.warc.gz"} |
What is the Higgs Boson?
What is this thing we keep hearing about – the Higgs Boson, and why is it important?
It’s been said that the best way to learn is to teach. And so, today I’m going to explain everything I can about the Higgs boson. And if I do this right, maybe, just maybe, I’ll understand it a
little better by the end of the episode.
I’d like to be clear that this video is for the person whose eyes glaze over every time you hear the term Higgs boson. You know it’s some kind of particle, Nobel prize, mass, blah blah. But you don’t
really get what it is and why it’s important.
First, let’s start with the Standard Model. These are essentially the laws of particle physics as scientists understand them. They explain all the matter and forces we see all around us. Well, most
of the matter, there are a few big mysteries, which we’ll discuss as we get deeper into this.
But the important thing to understand is that there are two major categories: the fermions and the bosons.
Bosons, fermions and other particles after a collsion. Credit: CERN
Fermions are matter. There are the protons and neutrons which are made up of quarks, and there are the leptons, which are indivisible, like electrons and neutrinos. With me so far? Everything you can
touch are these fermions.
The bosons are the particles that communicate the forces of the Universe. The one you’re probably familiar with is the photon, which communicates the electromagnetic force. Then there’s the gluon,
which communicates the strong nuclear force and the W and Z bosons which communicate the weak nuclear force.
Mystery number 1, gravity. Although it’s one of the fundamental forces of the Universe, nobody has discovered a boson particle that communicates this force. So, if you’re looking for a Nobel Prize,
find a gravity boson and it’s yours. Prove that gravity doesn’t have a boson, and you can also get a Nobel Prize. Either way, there’s a Nobel Prize in it for you.
Credit: PBS NOVA [1], Fermilab, Office of Science, United States Department of Energy, Particle Data Group
Again, this is the Standard Model, and it accurately describes the laws of nature as we see them around us.
One of the biggest unsolved mysteries in physics was the concept of mass. Why does anything have mass at all, or inertia? Why does the amount of physical “stuff” in an object define how easy it is to
get moving, or how hard it is to make it stop?
In the 1960s, physicist Peter Higgs predicted that there must be some kind of field that permeates all of space and interacts with matter, sort of like a fish swimming through water. The more mass an
object has, the more it interacts with this Higgs field.
And just like the other fundamental forces in the Universe, the Higgs field should have a corresponding boson to communicate the force – this is the Higgs boson.
The field itself is undetectable, but if you could somehow detect the corresponding Higgs particles, you could assume the existence of the field.
Cross-section of the Large Hadron Collider where its detectors are placed and collisions occur. LHC is as much as 175 meters (574 ft) below ground on the Frence-Swiss border near Geneva, Switzerland.
The accelerator ring is 27 km (17 miles) in circumference. (Photo Credit: CERN)
And this is where the Large Hadron Collider comes in. The job of a particle accelerator is to convert energy into matter, via the formula e=mc2. By accelerating particles – like protons – to huge
velocities, they give them an enormous amount of kinetic energy. In fact, in its current configuration, the LHC moves protons to 0.999999991c, which is about 10 km/h slower than the speed of light.
When beams of particles moving in opposite directions are crashed together, it concentrates an enormous amount of energy into a tiny volume of space. This energy needs somewhere to go so it freezes
out as matter (thanks Einstein). The more energy you can collide, the more massive particles you can create.
And so, in 2013, the LHC allowed physicists to finally be able to confirm the presence of the Higgs Boson by tuning the energy of the collisions to exactly the right level, and then detecting the
cascade of particles that occur when Higgs bosons decay.
Because the right particles are detected, you can assume the presence of the Higgs boson, and because of this, you can assume the presence of the Higgs field. Nobel prizes for everyone.
Particle collision. Credit: CERN
I said there were a few mysteries left; gravity was one, of course, but there are a few more. The reality is that physicists now know that the matter I described is really just a fraction of the
entire Universe. Cosmologists estimate that just 4% of the Universe is the normal baryonic matter that we’re familiar with.
Another 23% is dark matter, and a further 73% is dark energy. So there are still plenty of mysteries to keep physicists busy for years.
And so, in 2013, the Large Hadron Collider finally turned up the particle that physicists had predicted for 50 years. The last piece of the Standard Model was finally proven to exist, and we’re
closer to understanding what 4% of the Universe is. The other 96% (oh, and gravity), are still a total mystery.
Physicists are cranking up the LHC to higher and higher levels of energy, to search for other particles, to understand dark matter, and see if they can generate microscopic black holes. This mighty
instrument has plenty more science to reveal, so stay tuned.
That’s the Higgs Boson in a nutshell. Let me me know if there are other concepts in particle physics you’d like to talk about. Put your ideas into the comments below.
15 Replies to “What is the Higgs Boson?”
1. Now I get it. Although I’ll go one step further. Higgs Boson has no “particles’, no visible mass of its own. Its ‘mass’ (for lack of a better word) is intangible in all & any way, shape or form.
It exerts force proportionate to the mass sliding thru it. Sort of like a minnow and whale swimming in the same general area. However, as a mass propelles through the boson, the mass’s momentum
leaves a vortex behind that keeps pushing it, moving it in the same forward direction until (when and if) another force(s) bumps into it.
Nothing new here. It’s like geese flying in formation using the energy the geese in front leave behind.
From a human stand point This (Hypothetical) ‘Dark Matter’ will never be able to be measured or placed into any percentile of anything.
The wind can’t be seen but it can be felt. Dark Matter will NEVER be seen or be felt but we’re up to our necks in it. it isn’t made up of any particle of any size or polarity.
1. Scott, there is a fundamental difference between using c**2 as a mathematical constant used to calculate the magnitude of energy found in matter versus the actual, physical limit of speed
represented by the constant c. c**2 shows how energy greatly increases with mass, c does not change. Therefore, c**2 does not violate the value of c in any way. Hope that helps.
2. Interesting position, posted with much apparent confidence. A reply evokes too many questions to list: just two – 1. If the Higgs boson has no mass, was the mass of 127 GeV assigned only
because that was the calculated energy that manifested it? and 2. a.Why do you think the DM will never be seen, felt (non-anthropomorphically) or measured? b. Is it unattached subatomic
2. Thanks for the informative discussion about the Higgs. You are helping us novices understand what it is.
I have a general question about E=MC2. Hopefully you don’t mind me asking it here. Here goes: How can the Speed of Light be squared? I thought that a corollary of Einstein’s work is that the
Speed of Light is the Speed Limit; there is no going faster. However, he squares the Speed of Light in his famous equation. So, does this mean that the Speed of Light can be improved? Thanks.
1. Scott, there is a fundamental difference between using c**2 as a mathematical constant used to calculate the magnitude of energy found in matter versus the actual, physical limit of speed
represented by the constant c. c**2 shows how energy greatly increases with mass, c does not change. Therefore, c**2 does not violate the value of c in any way. Hope that helps.
1. Qedlin, thanks for the clarification. Now it makes more sense.
3. Your saying, the Higgs boson exists in the Higgs field that interacts with matter and thus gives mass to matter, must follow the scientific criterion of 21st century and not follow philosophical
and protoscientific practices of ancient world.
If protons entering the collision (LHC) got mass thanks to everywhere present Higgs bosons then Higgs bosons are not entering into the balance of mass and energy of that collision. After
collision, there must be equal amount mass and energy in products as in entry. Only, if they noticed more mass and energy in products then the scientific world may accept that Higgs boson was
found. Science is there where mass and energy cannot be created or destroyed, only modified in form.
If you are serious in proclaiming scientific truth then you must not omit at Higgs boson that they discovered really the part (the particle) having extreme density never before detected. This is
logical expectation of that collision since we know that extreme density of objects come to exist in crashes when they had a high speed. If we would consider the crash of objects (protons) having
a speed almost twice of the speed of light into a standing block then products of such a crash must have changed (deformed) their volumes.
Please, do not violate physics by witchcraft (materializing spirits and mathematics) and by pseudo-physics conclusions (creating particles by not respecting the balance of collisions).
4. Farewell to Higgs
5. Thank you for that “energy freezes out as matter” line. Just a retired high school chem/physics teacher here, so that idea of thinking of matter as precipitated energy clicks for me big time. Am
I correct in thinking the concept extends to nucleogenesis in supernovae? The energy is so high that matter essentially precipitates out as a low-volume form of energy?
6. What if we’re looking at gravity all wrong. We know it is intimately tied to mass somehow. We also know it is indistinguishable from acceleration. We also know that most of the mass of matter
comes from the binding energy within matter, with the final bit coming from the Higgs field. Also, tantalizing theoretical work done using Scale Symmetry theory has modeled the emergence of all
massed particles and their forces from the massless particles and their forces interacting alone. So what if gravity truly is only acceleration, with the Higgs field being the start of a cascade
to mass-fulness? We know that massless particles naturally travel at the speed of light, but no massed particles can ever reach these speeds. What if the mechanism of interaction between photons,
mesons, neutrinos that lead to the emergence of mass is the initial interactions with the Higgs fields, and gravity is akin to an inertial “impulse” impeded by the Higgs field, rather than
gravity being a particle and field configuration as everyone supposes? Perhaps we already have a unified field theory and no one knows it because we’ve assumed gravity must be a classical force
when in fact it is not…
1. I thought Einstein resolved the question with his general theory that gravity is not a force but a mass induced distortion of space time. Hence there is no need to search for a gravity boson.
There is none.
7. How can a single type of Higgs account for particles with different masses? Are some particles made up of multiple Higgs or do some Higgs have different masses? Are there any particles that have
less mass than the Higgs, if so how?
1. Different particle couple to the Higgs fields with different strengths– hence they have different masses. The Higgs field is constant, everywhere (except at LHC). It is not zero, but it has 0
Higgs Bosons: that is the so-called vacuum state. At LHC there is enough energy to excite the Higgs Field to a state with 1 Higgs Boson (it is massive because it also couples to the Higgs
An analogy would be the Electromagnetic Field: it’s vacuum state is 0 with 0 photons. The 1st excited state has 1 photon.
8. First of all – Thank you! An excellent and simple way to understand the Higgs Boson and the other terms often used in reference to it. Maybe you can help me….
I have a very serious question – I am not joking – regarding Einstein’s famous equation.
Either I am understanding something wrong or his iconic equation is not worth the paper it was written on. By definition:
In physics, mass–energy equivalence explains the relationship between mass and energy. It states every mass has an energy equivalent and vice versa—expressed using the formula
E = mc2
where E is the energy of a physical system, m is the mass of the system, and c is the speed of light in a vacuum (about 3×108 m/s). In words, energy equals mass multiplied by the speed of light
The way I understand this claim, then one gram of cotton candy has as much energy as one gram of lead. If I understand the definition of the equation properly, the cotton candy/lead equivalence
is not true.
If the “m” expressed the molecular/atomic bonding energy of the matter in question I would agree with it – as long as that expression truly reflected the level of molecular/atomic bonding energy.
Maybe I am missing something.
Any clarification would be greatly appreciated.
9. These particles are so small, so it cannot be hold in a container. I am curious if these particles can be redirected to rotate around a stripped neutrons
If it so happens to rotate we can add more of the same continuously so that they are in large numbers together, to know its properties | {"url":"https://www.universetoday.com/127086/what-is-the-higgs-boson/","timestamp":"2024-11-05T06:04:10Z","content_type":"text/html","content_length":"212431","record_id":"<urn:uuid:1dc175ac-3d2c-4eb3-9ad4-54fb70542ffc>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00319.warc.gz"} |
xpicture – Extensions of LaTeX picture drawing
The package extends the facilities of the pict2e and the curve2e packages, providing extra reference frames, conic section curves, graphs of elementary functions and other parametric curves.
Sources /macros/latex/contrib/xpicture
Version 1.2a
Licenses The LaTeX Project Public License 1.3
Copyright 2010, 2011, 2012 Robert Fuster
Maintainer Robert Fuster
TDS archive xpicture.tds.zip
Contained in TeXLive as xpicture
MiKTeX as xpicture
Topics Graphics in TeX
Curve Graphics
Download the contents of this package in one zip archive (1.3M).
Community Comments
Maybe you are interested in the following packages as well. | {"url":"https://ctan.org/pkg/xpicture","timestamp":"2024-11-08T06:01:08Z","content_type":"text/html","content_length":"16570","record_id":"<urn:uuid:df69483f-eaf5-4e66-9353-2cad805d98a4>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00048.warc.gz"} |
Search Results
In this lesson, students use an applet (technology tool) to hide a ladybug under a leaf. This requires experimentation, planning, and understanding of spatial relationships and visual memory.
This lesson engages students in creating a map of their hands. It provides purpose for using directional or positional words with mapping. The teacher draws a map of his or her hands and begins
mapping them using words the students suggest. This allows the teacher to assess positional concepts students currently know and to build on that knowledge. Students create a simple map.
In this lesson, students create a map of their face and practice locating different parts using the geometric and measurement concepts they have learned in previous lessons, including location,
navigation, spatial relationships, and measurement with nonstandard units. Students reproduce their face and describe it to reinforce their knowledge and skills of measuring and mapping. Using these
familiar territories connects mathematics with daily encounters.
Students count back to compare plates of fish-shaped crackers, and then they record the comparison in vertical and horizontal format. They apply their skills of reasoning and problem solving during
this lesson in several ways. [Because students have associated the word "more" with addition, the comparative approach to subtraction is typically more challenging for the students to understand.]
Students write subtraction problems, model them with sets of fish-shaped crackers, and communicate their findings in words and pictures. They record differences in words and in symbols. The additive
identity is reviewed in the context of comparing equal sets.
In this lesson, students determine differences using the number line to compare lengths. Because this meaning is based on linear measurement, it is a distinctly different representation from the
meanings presented in Lessons One and Two. At the end of the lesson, the students use reasoning and problem solving to predict differences and to answer puzzles involving subtraction.
This lesson encourages the students to explore another meaning for operations of subtraction, the balance. This meaning leads naturally into recording with equations. The students will imitate the
action of a pan balance and record the modeled subtraction facts in equation form.
In this lesson, the relation of addition to subtraction is explored with fish-shaped crackers. The students search for related addition and subtraction facts for a given number and also investigate
fact families when one addend or the difference is 0.
During this lesson, the students apply what they know about comparison subtraction by constructing bar graphs and using them to answer questions. They conduct a survey to gather data and then
complete a bar graph. They also use the data to generate a bar graph using technology.
During this final lesson in the unit, the students use the mathematical knowledge and skills developed in the previous lessons as they visit five stations to review comparative subtraction. | {"url":"https://illuminations.nctm.org/Search.aspx?view=search&type=ls&page=4","timestamp":"2024-11-05T22:27:36Z","content_type":"application/xhtml+xml","content_length":"68189","record_id":"<urn:uuid:885d40e6-5b72-48a6-9f21-591444dd1454>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00572.warc.gz"} |
This is part of the function module
This function measures the Pythagorean distance from a particular structure measured in the space defined by some set of collective variables.
This collective variable can be used to calculate something akin to:
\[ d(X,X') = \vert X - X' \vert \]
where \( X \) is the instantaneous values for a set of collective variables for the system and \( X' \) is the values that these self-same set of collective variables take in some reference structure
provided as input. If we call our set of collective variables \(\{s_i\}\) then this CV computes:
\[ d = \sqrt{ \sum_{i=1}^N (s_i - s_i^{(ref)})^2 } \]
where \(s_i^{(ref)}\) are the values of the CVs in the reference structure and \(N\) is the number of input CVs.
We can also calculate normalized euclidean differences using this action and the METRIC=NORM-EUCLIDEAN flag. In other words, we can compute:
\[ d = \sqrt{ \sum_{i=1}^N \sigma_i (s_i - s_i^{(ref)})^2 } \]
where \(\sigma_i\) is a vector of weights. Lastly, by using the METRIC=MAHALONOBIS we can compute Mahalonobis distances using:
\[ d = \left( \mathbf{s} - \mathbf{s}^{(ref)} \right)^T \mathbf{\Sigma} \left( \mathbf{s} - \mathbf{s}^{(ref)} \right) \]
where \(\mathbf{s}\) is a column vector containing the values of all the CVs and \(\mathbf{s}^{(ref)}\) is a column vector containing the values of the CVs in the reference configuration. \(\mathbf{\
Sigma}\) is then an \(N \times N\) matrix that is specified in the input.
The following input calculates the distance between a reference configuration and the instantaneous position of the system in the trajectory. The position of the reference configuration is specified
by providing the values of the distance between atoms 1 and 2 and atoms 3 and 4.
Click on the labels of the actions for more information on what each action computes
d1: DISTANCE =1,2 The DISTANCE action with label d1 calculates a single scalar value
d2: DISTANCE =3,4 The DISTANCE action with label d2 calculates a single scalar value
t1: TARGET =reference.pdb =EUCLIDEAN The TARGET action with label t1 calculates a single scalar value
PRINT =t1 =colvar The PRINT action with label
The contents of the file containing the reference structure (reference.pdb) is shown below. As you can see you must provide information on the labels of the CVs that are being used to define the
position of the reference configuration in this file together with the values that these quantities take in the reference configuration.
DESCRIPTION: a reference point.
REMARK WEIGHT=1.0
REMARK ARG=d1,d2
REMARK d1=1.0 d2=1.0
Glossary of keywords and components
Compulsory keywords
TYPE ( default=EUCLIDEAN ) the manner in which the distance should be calculated
a file in pdb format containing the reference structure. In the PDB file the atomic coordinates and box lengths should be in Angstroms unless you are working with natural units. If you are
REFERENCE working with natural units then the coordinates should be in your natural length unit. The charges and masses of the atoms (if required) should be inserted in the beta and occupancy
columns respectively. For more details on the PDB file format visit http://www.wwpdb.org/docs.html
NUMERICAL_DERIVATIVES ( default=off ) calculate the derivatives for these quantities numerically | {"url":"https://www.plumed.org/doc-v2.8/user-doc/html/_t_a_r_g_e_t.html","timestamp":"2024-11-12T22:26:13Z","content_type":"application/xhtml+xml","content_length":"12748","record_id":"<urn:uuid:86bf8e6b-c691-4856-8190-0420e25fea92>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00008.warc.gz"} |
An Innovative Method to Improve Model Accuracy by Implementing Multi-models Scheme for 28nm Node and Below
Abstract: As the process comes into 28nm node and below, lithography struggles stronger between high resolution (high NA) and enough process window especially for hole layers (Contacts and Vias).
Taking more care of process window may result in lower image quality of structures and bigger uncertainty in OPC model accuracy. Besides, it is difficult to cover all kinds of test structures within
acceptable accuracy in one OPC model because of distinct difference of image quality of different patterns. To solve these problems, this paper introduces an innovative method of applying
multi-models in one layer OPC. According to different characteristic features, multiple models are applied respectively and the fitting on these features with poor resolution can be improved by
re-optimizing based on related model. A practice for 28 nm Via layer modeling calibration is given, and it shows an evident improvement of model accuracy through the implementing of multiple models
Keywords: Image quality; lithography; OPC model; multi-model | {"url":"http://jommpublish.org/p/35/","timestamp":"2024-11-05T00:02:57Z","content_type":"text/html","content_length":"60530","record_id":"<urn:uuid:ee471f8f-3435-47d3-b2bb-ba07a15a988d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00438.warc.gz"} |
Halfband Filter
FIR Halfband Filter Design
This example shows how to design FIR halfband filters. Halfband filters are widely used in multirate signal processing applications when you interpolate or decimate by a factor of two. In many cases,
you can implement halfband filters efficiently in a polyphase form because nearly half of the halfband filter coefficients are equal to zero.
Halfband filters have two important characteristics:
• The passband and stopband ripples must be the same.
• The passband- and stopband-edge frequencies are equidistant from the halfband frequency $\frac{\mathrm{Fs}}{4}$ Hz or $\frac{\pi }{2}$ rad/sample in normalized frequency.
Design a Halfband Filter using the designHalfbandFIR function
Use the designHalfbandFIR function to design an FIR halfband filter. The function returns the filter coefficients or one of the supported FIR filter objects.
Design an Equiripple Halfband Filter
Consider a halfband filter operating on data sampled at 96 kHz with a passband frequency edge of 22 kHz. The halfband frequency corresponding to that sample rate is 24 kHz, which amounts to a quarter
of the sample rate. The transition width of that design is 2kHz, or 1/12 rad/sec in normalized units. Set DesignMethod to "equiripple".
Fs = 96e3; % Sample rate
Fp = 22e3; % Passband edge
Fh = Fs/4; % Halfband frequency
TW = (Fh-Fp)/Fh; % Normalized transition width
N = 100; % Filter order
num = designHalfbandFIR(FilterOrder=N,TransitionWidth=TW,Passband="lowpass",DesignMethod="equiripple");
By zooming in on the response, you can verify that the passband and stopband peak-to-peak ripples are the same. There is symmetry about the $\frac{\mathrm{Fs}}{4}$ (24 kHz) point. The passband
extends up to 22 kHz as specified and the stopband begins at 26 kHz. You can also verify that every other coefficient is equal to zero by looking at the impulse response. This makes the filter very
efficient to implement for interpolation or decimation by a factor of 2.
Filtering Streaming Data Through a Halfband Filter
When working with streaming data, use the dsp.FIRFilter, dsp.FIRHalfbandInterpolator and dsp.FIRHalfbandDecimator System objects, which provide efficient filter implementations. These System objects
support filtering double and single precision floating-point data as well as fixed-point data. They also support C and HDL code generation as well as optimized ARM® Cortex® M and ARM® Cortex® A code
generation. When using dsp.FIRFilter, use designHalfbandFIR with Structure set to 'single-rate'. When using dsp.FIRHalfbandInterpolator and dsp.FIRHalfbandDecimator, set Structure to 'interp' or
'decim' respectively.
Construct an FIR halfband single-rate filter.
num = designHalfbandFIR(FilterOrder=N,TransitionWidth=TW,Passband="lowpass",DesignMethod="equiripple",Structure="single-rate");
halfbandFilter = dsp.FIRFilter(num)
halfbandFilter =
dsp.FIRFilter with properties:
Structure: 'Direct form'
NumeratorSource: 'Property'
Numerator: [0 2.1118e-04 0 -2.4012e-04 0 3.7199e-04 0 -5.4746e-04 0 7.7546e-04 0 -0.0011 0 0.0014 0 -0.0019 0 0.0024 0 -0.0031 0 0.0039 0 -0.0049 0 0.0060 0 -0.0074 0 0.0090 0 -0.0110 0 0.0134 0 -0.0163 0 0.0201 0 -0.0252 0 … ] (1×101 double)
InitialConditions: 0
Show all properties
You can create a System object directly through designHalfbandFIR by setting SystemObject parameter to true.
Construct an FIR halfband interpolator.
halfbandInterpolator = designHalfbandFIR(FilterOrder=N, TransitionWidth=TW,...
Self-Design Mode of dsp.FIRHalfbandInterpolator and dsp.FIRHalfbandDecimator
The dsp.FIRHalfbandInterpolator and dsp.FIRHalfbandDecimator System objects can internally design the filter coefficients. To use that option, construct a dsp.FIRHalfbandInterpolator or a
dsp.FIRHalfbandDecimator System object, and specify which design specification is used through the Specification property. The object is self-designing when Specification is not set to
'Coefficients'. The objects use the same parameter names as the designHalfbandFIR function: FilterOrder, TransitionWidth, and StopbandAttenuation. Another advantage of using self-designing mode the
support for arbitrary sample rates in addition to normalize frequency units.
Construct a halfband interpolator at a sample rate of 96 kHz, with filter order of 50, and a transition width of 4 kHz.
Fs = 96e3; % Sample rate
TW = 4e3; % Transition width
N = 50; % Filter order
halfbandInterpolator = dsp.FIRHalfbandInterpolator(SampleRate=Fs,...
Specification="Filter order and transition width",...
As long as the object is not locked, you can tune the design parameters, and the design changes accordingly.
Use the tf function to extract the coefficients of the FIR halfband interpolator or decimator System object. The filter order in the example above is 100, as expected.
num = tf(halfbandInterpolator);
To perform the halfband interpolation, pass the input data through the dsp.FIRHalfbandInterpolator System object.
FrameSize = 256;
sine1 = dsp.SineWave(Frequency=10e3,SampleRate=Fs,SamplesPerFrame=FrameSize);
sine2 = dsp.SineWave(Frequency=20e3,SampleRate=Fs,SamplesPerFrame=FrameSize);
x = sine1() + sine2() + 0.01.*randn(FrameSize,1); % Input signal
y = halfbandInterpolator(x); % Step through the object
Plot the interpolated samples overlaid on the input samples by compensating for the delay of the filter. Use the outputDelay function to obtain the filter delay. The input samples remain unchanged at
the output of the filter because one of the polyphase branches of the halfband filter is a pure delay branch that does not change the input samples.
[D, FsOut] = halfbandInterpolator.outputDelay();
nx = 0:length(x)-1;
ny = 0:length(y)-1;
hold on
hold off
legend("Input samples","Interpolated samples")
xlim([1e-3, 1.4e-3])
Plot the spectral content of the output signal using spectrumAnalyzer. In the case of interpolation, you upsample by a factor of 2 and then filter (conceptually), therefore you need to specify the
sample rate in spectrumAnalyzer as $2\mathrm{Fs}$ because of the upsampling by 2.
scope = spectrumAnalyzer(SampleRate=2*Fs);
while toc < 10
x = sine1() + sine2() + 0.01.*randn(FrameSize,1); % 96 kHz
y = halfbandInterpolator(x); % 192 kHz
The spectral replicas are attenuated by about 40 dB, which is roughly the attenuation provided by the halfband filter.
In the case of decimation, the sample rate you specify in the dsp.FIRHalfbandDecimator corresponds to the sample rate of the input, since the object filters and then downsamples (conceptually).
Cascade a halfband decimator with the halfband interpolator that you designed in the previous section, and plot the spectral content of the output. The sample rate at the output of the two blocks is
Fs, just like the input.
halfbandDecimator = dsp.FIRHalfbandDecimator(SampleRate=2*Fs,...
Specification="Filter order and transition width",...
scope = spectrumAnalyzer(SampleRate=Fs);
while toc < 10
x = sine1() + sine2() + 0.01.*randn(FrameSize,1); % 96 kHz
y = halfbandInterpolator(x); % 192 kHz
xd = halfbandDecimator(y); % 96 kHz again
Minimum Order Design Specifications
Instead of specifying the filter order, you can specify a transition width and a stopband attenuation. The halfband filter design algorithm automatically attempts to find the smallest filter order
which satisfies the specifications.
Ast = 80; % 80 dB
num = designHalfbandFIR(StopbandAttenuation=Ast,TransitionWidth=4000/Fs,DesignMethod="equiripple");
minimumOrder = length(num)
The same can be done with the dsp.FIRHalfbandInterpolator and the dsp.FIRHalfbandDecimator objects. However, you need to explicitly set the Specification property to "Transition width and stopband
halfbandInterpolator = dsp.FIRHalfbandInterpolator(SampleRate=Fs,Specification="Transition width and stopband attenuation",...
StopbandAttenuation=Ast, TransitionWidth=4000);
Specify Filter Order and Stopband Attenuation
You can also design the filter by specifying the filter order and the stopband attenuation.
num = designHalfbandFIR(StopbandAttenuation=Ast,FilterOrder=N,DesignMethod="equiripple");
The same can be done with the dsp.FIRHalfbandInterpolator and the dsp.FIRHalfbandDecimator objects. However, you need to explicitly set the Specification property to "Filter order and stopband
halfbandDecimator = dsp.FIRHalfbandDecimator(SampleRate=Fs,...
Specification="Filter order and stopband attenuation",...
Use Halfband Filters for Filter Banks
You can use halfband interpolators and decimators to efficiently implement synthesis and analysis filter banks. The halfband filters discussed so far in the example were all lowpass filters. With a
single extra adder, you can obtain a highpass response in addition to the lowpass response and use the two responses to implement the filter bank.
Simulate a quadrature mirror filter (QMF) bank. First, separate an 8 kHz signal consisting of 1 kHz and 3 kHz sine waves into two half-rate signals (4 kHz) using a lowpass and highpass halfband
decimator. The lowpass signal retains the 1 kHz sine wave while the highpass signal retains the 3 kHz sine wave (which is aliased to 1 kHz after downsampling). Merge the signals back together with a
synthesis filter bank using a halfband interpolator. The highpass branch upconverts the aliased 1 kHz sine wave back to 3 kHz. The interpolated signal has an 8 kHz sample rate.
Fs1 = 8000; % Units = Hz
Spec = "Filter order and transition width";
Order = 52;
TW = 4.1e2; % Units = Hz
% Construct FIR Halfband Interpolator
halfbandInterpolator = dsp.FIRHalfbandInterpolator( ...
% Construct FIR Halfband Decimator
halfbandDecimator = dsp.FIRHalfbandDecimator( ...
% Input
f1 = 1000;
f2 = 3000;
InputWave = dsp.SineWave(Frequency=[f1,f2],SampleRate=Fs1,...
SamplesPerFrame=1024,Amplitude=[1 0.25]);
% Construct Spectrum Analyzer object to view the input and output
scope = spectrumAnalyzer(SampleRate=Fs1,...
YLimits=[-120 30],...
Title="Input Signal and Output Signal of Quadrature Mirror Filter",...
while toc < 10
Input = sum(InputWave(),2);
NoisyInput = Input+(10^-5)*randn(1024,1);
[Lowpass,Highpass] = halfbandDecimator(NoisyInput);
Output = halfbandInterpolator(Lowpass,Highpass);
Kaiser Window Designs
All designs presented so far in the example were optimal equiripple designs. The designHalfbandFIR function, as well as the dsp.FIRHalfbandDecimator and dsp.FIRHalfbandInterpolator System objects can
also design halfband filters using the Kaiser window method.
Compare an equiripple and Kaiser window filter designs. Use the same specifications sample rate, filter order, and transition bandwidth that were used in the previous steps.
Fs = 44.1e3;
N = 90;
TW = 1000;
equirippleHBFilter = designHalfbandFIR(DesignMethod="equiripple",...
kaiserHBFilter = designHalfbandFIR(DesignMethod="kaiser",...
Compare the designs using filterAnalyzer. The two designs allow for tradeoffs between minimum stopband attenuation and larger overall attenuation.
FA = filterAnalyzer({equirippleHBFilter,1,1},{kaiserHBFilter,1,1},SampleRates=2*Fs);
setLegendStrings(FA,["Equiripple design","Kaiser-window design"])
Specify the Stopband Attenuation in Kaiser Design
Alternatively, one can specify the order and the stopband attenuation. This allows for tradeoffs between overall stopband attenuation and transition width.
Ast = 60; % Minimum stopband attenuation
equirippleHBFilter = designHalfbandFIR(DesignMethod="equiripple",...
kaiserHBFilter = designHalfbandFIR(DesignMethod="kaiser",...
FA = filterAnalyzer({equirippleHBFilter,1,1},{kaiserHBFilter,1,1},SampleRates=2*Fs);
setLegendStrings(FA,["Equiripple design","Kaiser-window design"])
Minimum-Order Kaiser Designs
The Kaiser window design method also supports minimum order designs. Specify a target transition width and a stopband attenuation, and the design algorithm attempts to find the smaller order Kaiser
FIR halfband that satisfies the target specifications. A minimum-order Kaiser design usually has a larger order than its equiripple counterpart, but the overall stopband attenuation is better in
Fs = 44.1e3;
TW = 1000; % Transition width
Ast = 60; % 60 dB minimum attenuation in the stopband
equirippleHBFilter = designHalfbandFIR(DesignMethod="equiripple",...
kaiserHBFilter = designHalfbandFIR(DesignMethod="kaiser",...
FA = filterAnalyzer({equirippleHBFilter,1,1},{kaiserHBFilter,1,1}, SampleRates=Fs);
setLegendStrings(FA,["Equiripple design","Kaiser-window design"])
Automatically Set Filter Design Method
The Kaiser method and Equiripple method have different strengths depending on the design specifications. For tight design specifications such as very high stopband attenuation or a very narrow
transition width, the Equiripple design method often fails to converge. In such cases, the Kaiser method is superior.
This code illustrates a case where the filter specifications are too tight to use an equiripple design. Set the DesignMethod property to "Equiripple", and the design fails to converge as you can
observe from the frequency response of the filter.
TW = 0.001;
Ast = 180; % 180 dB minimum attenuation in the stopband
equirippleHBFilter = designHalfbandFIR(DesignMethod="equiripple",...
FA = filterAnalyzer(equirippleHBFilter);
setLegendStrings(FA,"DesignMethod = equiripple");
Repeat the design with DesignMethod set to "kaiser". While the kaiser-based filter design does not accurately meet the tight design specifications, the filter better converges as you can see by
comparing the frequency response of the two filters.
kaiserHBFilter = designHalfbandFIR(DesignMethod="kaiser",...
FA = filterAnalyzer(kaiserHBFilter);
setLegendStrings(FA,"DesignMethod = kaiser");
In addition to "equiripple" and "kaiser", you can set DesignMethod to "Auto". When you set the DesignMethod to "Auto", the designHalfbandFIR function selects the design method automatically based on
the filter design parameters. This is the default design method used when you don't specify the DesignMethod parameter. You can determine which design method is selected by passing the Verbose=true
argument. In the code below, the function automatically selects an Equiripple design.
TW = 1/44.1;
Ast = 180;
autoHBFilter = designHalfbandFIR(DesignMethod="auto",...
StopbandAttenuation=Ast, ...
designHalfbandFIR(TransitionWidth=0.022675736961451247, StopbandAttenuation=180, DesignMethod="kaiser", Passband="lowpass", Structure="single-rate", SystemObject=false)
FA = filterAnalyzer(autoHBFilter);
setLegendStrings(FA,"designHalfbandFIR, DesignMethod = auto");
The dsp.FIRHalfbandDecimator and dsp.FIRHalfbandInterpolator System objects also support the "auto" design method.
Fs = 44.1e3;
TW = 1000; % Transition width
Ast = 60; % 60 dB minimum attenuation in the stopband
autoHBFilter = dsp.FIRHalfbandDecimator(Specification="Transition width and stopband attenuation",...
StopbandAttenuation=Ast, ...
FA = filterAnalyzer(autoHBFilter);
setLegendStrings(FA,"dsp.FIRHalfbandDecimator, DesignMethod = auto");
If the design constraints are very tight, then the design algorithm automatically selects the Kaiser window method, as this method proves to be optimal choice when designing filters with very tight
specifications. However, if the design constraints are not tight, then the algorithm selects the equiripple design method. | {"url":"https://ch.mathworks.com/help/dsp/ug/fir-halfband-filter-design.html","timestamp":"2024-11-03T16:59:44Z","content_type":"text/html","content_length":"103149","record_id":"<urn:uuid:3408069d-0e16-4def-8955-1dde5e526217>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00249.warc.gz"} |
Propagation of nonlinearities in the inertia matrix of tracked vehicles
In this investigation, a procedure is presented for the numerical solution of tracked vehicle dynamics equations of motion. Tracked vehicles can be represented as two kinematically decoupled
subsystems. The first is the chasis subsystem which consists of chasis, rollers, idlers, and sprockets. The second is the track subsystem which consists of track links interconnected by revolute
joints. While there is dynamic force coupling between these two subsystems, there is no inertia coupling since the kinematic equations of the two subsystems are not coupled. The objective of the
procedure developed in this investigation is to take advantage of the fact that in many applications, the shape of the track does not significantly change even though the track links undergo
significant configurations changes. In such cases the nonlinearities propagate along the diagonals of a velocity influence coefficient matrix. This matrix is the only source of nonlinearities in the
generalized inertia matrix. A permutation matrix is introduced to minimize the number of generalized inertia matrix LU factor evaluations for the track.
Conference Proceedings of the 1994 ASME Design Technical Conferences. Part 1 (of 2)
City Minneapolis, MN, USA
Period 11/09/94 → 14/09/94
Dive into the research topics of 'Propagation of nonlinearities in the inertia matrix of tracked vehicles'. Together they form a unique fingerprint. | {"url":"https://khu.elsevierpure.com/en/publications/propagation-of-nonlinearities-in-the-inertia-matrix-of-tracked-ve-2","timestamp":"2024-11-10T04:13:22Z","content_type":"text/html","content_length":"52952","record_id":"<urn:uuid:e81c7945-8747-46fc-9e89-554766a5de0d>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00295.warc.gz"} |
Neural Network Normalization - Get Neural Net
Neural Network Normalization
Neural networks are powerful machine learning models that can be used for various tasks such as image recognition, natural language processing, and predicting future outcomes. However, the success of
a neural network largely depends on the quality of its training data. One crucial step in preparing the data for a neural network is normalization. In this article, we will explore what normalization
is, why it is important, and different methods of normalization to improve the performance and efficiency of neural networks.
Key Takeaways:
• Normalization is a data preprocessing technique used to standardize the scale and range of input features.
• Normalizing input data improves the convergence speed and performance of neural networks.
• Popular normalization techniques include min-max scaling, z-score normalization, and feature scaling.
• Normalization should be applied to both input features and output values in regression tasks.
• Batch normalization is a technique used to normalize the activations of intermediate layers in a neural network.
What is Normalization?
In the context of neural networks, normalization refers to the process of transforming input data so that it conforms to a specific range or distribution. The main objective is to bring all input
features within a similar scale, which helps the network to learn effectively and prevents certain features from dominating the learning process.
For example, if we have a neural network that takes two input features, “age” (ranging from 0 to 100) and “salary” (ranging from 10000 to 100000), the difference in scale between these two features
can cause issues during training. Normalization can be used to scale both features to a common range, such as 0 to 1 or -1 to 1, ensuring that both inputs are equally important in the learning
**Normalization plays a crucial role in improving the performance and efficiency of neural networks.** By bringing input features to a similar scale, neural networks can reach convergence faster and
make more accurate predictions.
Common Normalization Techniques
There are several widely used normalization techniques that can be applied to neural networks, depending on the nature of the data and the task at hand. These techniques ensure that the data is
transformed in a way that is both meaningful and doesn’t introduce any bias into the learning process. Let’s explore some of these techniques:
1. Min-Max Scaling
The min-max scaling, also known as feature scaling, is a popular normalization technique. It transforms the data to a specific range, usually between 0 and 1, by subtracting the minimum value and
dividing by the range (maximum value minus the minimum value).
• Benefits:
□ Preserves the original distribution of the data.
□ Useful for algorithms that assume the data is normally distributed.
• Drawbacks:
□ Sensitive to outliers in the data.
2. Z-Score Normalization
Z-score normalization, also known as standardization, transforms the data so that it has a mean of 0 and a standard deviation of 1. This technique is often used when the data does not follow a normal
• Benefits:
□ Handles outliers well.
□ Works well with algorithms that assume the data has zero-mean and equal variance.
• Drawbacks:
□ Does not preserve the original distribution of the data.
Batch Normalization
Batch normalization is a normalization technique applied to the activations of intermediate layers in a neural network. It aims to normalize the mean and variance of the layer’s inputs, making the
network more robust and less sensitive to changes in the input distribution.
**One interesting benefit of batch normalization is that it acts as a regularizer, reducing the need for other regularization techniques such as dropout.** It can also speed up the training process
by allowing higher learning rates.
Normalization in Regression Tasks
Normalization is not only important for the input features but also for the output values in regression tasks. If the output values have a large range, it can lead to slow convergence and instability
during training. Applying normalization techniques to the output values can help to stabilize the learning process.
Tables and Data Points
Normalization Technique Benefits Drawbacks
Min-Max Scaling Preserves original distribution Sensitive to outliers
Useful for normally distributed data
Z-Score Normalization Handles outliers well Does not preserve original distribution
Works well with specific assumptions
Normalization is an essential step in preparing data for neural networks. It ensures that input features are scaled and distributed in a way that improves the network’s performance and convergence.
Different normalization techniques, such as min-max scaling and z-score normalization, can be applied depending on the data characteristics and the learning task at hand. Additionally, batch
normalization can be used to normalize intermediate layer activations and improve the robustness of the network. By normalizing both input features and output values, neural networks can achieve
better results and more efficient training.
Common Misconceptions
Misconception 1: Normalization is not necessary for Neural Networks
One common misconception about neural networks is that normalization is not necessary and does not have a significant impact on the overall performance. However, normalization is crucial for neural
networks in order to achieve better convergence and prevent features from dominating the learning process.
• Normalization helps avoid bias towards features with larger scales.
• Normalizing the data can accelerate the learning process by reducing the number of iterations needed for convergence.
• Normalization improves the generalization ability of the neural network.
Misconception 2: Normalization can be applied equally to all types of data
Another misconception is the belief that normalization can be applied equally to all types of data. In reality, different types of data require different normalization techniques. For instance, time
series data requires normalization based on the time interval, while image data may need normalization based on pixel intensities.
• Different normalization techniques include min-max scaling, z-score normalization, and decimal scaling.
• Normalizing time series data often involves techniques like zero-mean normalization or scaling by the standard deviation.
• Image data normalization can involve techniques such as contrast stretching or histogram equalization.
Misconception 3: Normalization guarantees improved performance in all cases
One misconception is that normalization guarantees improved performance in all cases. While normalization is generally beneficial, there are situations where it may not lead to significant
improvements, or even in some cases, hinder the performance of the neural network.
• Normalization in some cases may introduce noise or distort the distribution of the data.
• When the data is already well-distributed and has similar scales, normalization may not have a noticeable impact.
• The choice of normalization technique and parameters can also affect the performance of the neural network.
Misconception 4: Normalization is only about scaling the input data
Another common misconception is that normalization is solely about scaling the input data to a specific range. While scaling is a crucial part of normalization, it is not the only aspect.
Normalization can also involve other transformations, such as handling missing values, dealing with outliers, or transforming the data to follow a specific distribution.
• Normalization can involve removing outliers by clipping or winsorizing the data.
• Imputing missing values using techniques such as mean substitution or regression imputation is also part of normalization.
• Transforming the data to a specific distribution, such as log-transforming skewed data, can be part of the normalization process.
Misconception 5: Normalization is a one-time process
A common misconception is that normalization is a one-time process that is applied before training the neural network. In reality, normalization is often an iterative process that needs to be
performed at multiple stages, including during training, testing, and evaluation, in order to achieve optimal performance.
• Normalization needs to be applied to both the input features and the target labels.
• During training, normalization helps ensure stable learning and prevent divergence.
• Normalization during testing and evaluation is necessary to maintain consistency and ensure fair comparison of results.
Normalization Techniques in Neural Networks
Normalization is an essential technique used in neural networks to bring data into a standard format, making it more suitable for processing. Different normalization methods have been developed to
enhance the performance and accuracy of neural networks. In this article, we explore ten different normalization techniques and their potential impact on neural network models.
Table: Min-Max Normalization
The Min-Max normalization technique scales data within a specific range. This table showcases the application of Min-Max normalization to a sample dataset of temperatures in degrees Celsius:
City Temperature (°C) Normalized Value
London 15 0.375
New York 22 0.625
Tokyo 30 1
Table: Standardization
Standardization centers the data around 0 with a standard deviation of 1. The table below demonstrates the effect of standardization on a dataset of students’ test scores:
Student Test Score Standardized Score
John 85 0.47
Sarah 90 0.77
Michael 70 -0.93
Table: Z-Score Normalization
Z-Score normalization transforms the data into values that represent how many standard deviations an observation is from the mean. The table presents the Z-Score normalized ages of a population:
Individual Age Z-Score
Person 1 30 0.44
Person 2 45 1.47
Person 3 20 -0.53
Table: Robust Scaling
Robust scaling normalizes the data using statistics that are more resilient to outliers. The table below showcases the effect of robust scaling on a dataset of household incomes:
Household Income ($) Scaled Income
Household 1 40,000 -0.67
Household 2 80,000 0.33
Household 3 500,000 3.33
Table: Unit Vector Scaling
Unit vector scaling normalizes data by dividing each value by the length of the vector formed by the observations. The table depicts the unit vector scaled dimensions of various objects:
Object Dimension 1 Dimension 2 Normalized Dimension 1 Normalized Dimension 2
Chair 2 3 0.55 0.83
Table 5 8 0.51 0.85
Bookshelf 10 4 0.89 0.45
Table: Log Transformation
Log transformation applies the natural logarithm function to the data. This table displays the effect of log transformation on a dataset of product prices:
Product Price ($) Log Price
Product A 50 3.91
Product B 200 5.3
Product C 1000 6.91
Table: Quantile Transformation
Quantile transformation maps the data to a specified distribution, often a normal distribution. The table illustrates the quantile-transformed heights of a sample population:
Individual Height (cm) Transformed Height
Individual 1 150 -2.33
Individual 2 170 -0.67
Individual 3 180 1.18
Table: Power Transformation
Power transformation raises the data to a specific power, providing a way to adjust skewed distributions. The table demonstrates the effect of a square root transformation on a dataset of rainfall
amounts (in mm):
City Rainfall (mm) Transformed Rainfall
City 1 16 4
City 2 81 9
City 3 144 12
Table: Feature Scaling
Feature scaling normalizes each feature independently, enabling fair comparisons across different attributes. The table showcases feature scaling on a dataset of speed limits (in mph):
Road Type Speed Limit (mph) Scaled Speed Limit
Residential 30 0.25
Highway 65 0.54
Rural 55 0.46
Normalization techniques play a vital role in preparing data for neural network training, improving model performance, and achieving accurate predictions. The choice of normalization method depends
on the characteristics and requirements of the data used in a given neural network model.
Frequently Asked Questions
What is normalization in neural networks?
Normalization in neural networks refers to the process of scaling the input data to a standard range to ensure more efficient and effective training. It helps prevent any feature from dominating the
learning process due to its larger values and allows the algorithm to converge faster.
Why is normalization important in neural networks?
Normalization is essential in neural networks to avoid biases towards certain input features that may have different scales or units. By normalizing the data, the network can treat all features
equally, leading to better generalization and improved performance.
What are the different normalization techniques used in neural networks?
There are several normalization techniques used in neural networks, including min-max scaling, z-score normalization, decimal scaling, and softmax normalization. Each technique has its own advantages
and is suitable for specific scenarios.
How does min-max scaling work?
Min-max scaling (also known as normalization) rescales the data to a specific range, typically between 0 and 1. It subtracts the minimum value from each data point and divides it by the range
(maximum value – minimum value) of the feature.
What is z-score normalization?
Z-score normalization (standardization) transforms the data so that it has a mean of 0 and a standard deviation of 1. It subtracts the mean from each data point and divides it by the standard
deviation of the feature.
Explain decimal scaling normalization.
Decimal scaling normalization involves scaling the data by dividing each value by the maximum absolute value of the feature. The resulting values lie between -1 and 1, preserving the order of
magnitude of the original data.
What is softmax normalization?
Softmax normalization computes the exponential value of each data point and then divides it by the sum of the exponential values of all data points in the same feature. This type of normalization
converts the values into probabilities that sum up to 1.
Can normalization be applied to both input features and output labels?
Yes, normalization can be applied to both input features and output labels in neural networks. It is important to ensure that the entire data, including inputs and outputs, is normalized consistently
to achieve accurate and reliable predictions.
When should normalization be performed in the neural network workflow?
Normalization should typically be performed after the data preprocessing step and before training the neural network. This ensures that the data is properly normalized and ready for the learning
algorithm to process and learn from.
Are there any cases where normalization might not be necessary?
In some cases, normalization may not be necessary, especially when all input features are already on a similar scale or when the chosen machine learning algorithm is known to be robust to varying
feature scales. However, it is generally recommended to apply normalization to improve the overall performance and stability of the neural network. | {"url":"https://getneuralnet.com/neural-network-normalization/","timestamp":"2024-11-10T05:57:26Z","content_type":"text/html","content_length":"65898","record_id":"<urn:uuid:c92f1289-e2da-45d5-90ad-6aea5ee960fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00699.warc.gz"} |
Percentage Calculator
Xoom Free Percentage Calculator Tool
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and are available in some states with income-based tax credit options as well (also see here). State or local insurance companies may offer one of these tools through their website when a
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Calculate basic and complex percentages with our free online % calculator! What is X of Y? Despite its simplicity, the tool is easy to use and learn in a matter of minutes. Two fields must be filled
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Using our tool is as simple as inputting different combinations in different areas. For simple tracking, it also gives you the entire history of all your computations.
Why do we use the term percentage?
As a number, ratio, or fraction may be stated in many different ways, a percentage is only one of them. We use it all the time to describe probabilities and scores.
‘Percent’ is the symbol used to represent it. 10 percent, for example, might represent 0.01, 10 percent, or ten hundredths.
This is not a mistake if you’ve seen the sign with additional circles. What that implies is this:
(per Millie) – ten thousand (basis point)
Uses of Percentage Calculator:
Percentage Calculator: Uses and Applications
Financial management relies on these activities in nearly every area of life.
– Coupons and posters proclaiming “discounts” are common sights at grocery stores, as are bills with complex percentage computations.
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Real and Imaginary Numbers - FilipiKnow
Real and Imaginary Numbers
Numbers are everywhere!
The money in your pocket, your birthdate, class grades, and even the time you will spend reading this reviewer. All of these things involve numbers.
Although we are all familiar with numbers, not everyone knows their classifications.
In this review, let’s discuss two classes of numbers: Real and imaginary numbers.
Click below to go to the main reviewers:
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Table of Contents
Natural or Counting Numbers
Natural numbers are used to count objects. These are the numbers you use to determine how many pets you have, how many apples you have bought from the market, or how many petals a flower has.
Natural numbers start with 1, followed by 2, then 3, and so on. Therefore, the smallest natural number is 1. Meanwhile, there’s no largest natural number.
0, negative numbers, fractions, and decimals are not natural numbers since we do not use them to count objects.
Numbers such as 14, 25, 799, 1212, 100000, and 5612312 are all examples of natural numbers.
Whole Numbers
Whole numbers are counting numbers, including 0. In other words, if you include 0 in the set of natural numbers, you will obtain the set of whole numbers.
Therefore, natural numbers are part of the set of whole numbers.
This means that all natural numbers are also whole numbers. Also, all whole numbers are natural numbers except for 0 since 0 is not a natural number.
Example: Is 510 a whole number or a natural number?
Solution: 510 is a natural number since we use it for counting. Since all natural numbers are also whole numbers, 510 is both a natural and a whole number.
Integers are positive and negative whole numbers, including 0. Examples are 1, 0, -1, 500, -500, 2200, -3800, and so on.
If you combine the set of all whole numbers and the set of all whole negative numbers, you now have the set of integers. Also, the set of natural numbers and the set of whole numbers are just subsets
or parts of the set of integers.
Fractions and decimals are not integers. For example, ½, -⅓, -⅔, and 0.9 are not integers.
Rational Numbers
Any number that can be expressed as a ratio of two integers is a rational number. In simple words, a number is a rational number if there are two integers such that when the first integer is divided
by the second integer, the result is the original number.
Suppose the number 18. Can you think of two integers such that when you divide the first integer by the second integer, the result is 18?
36 and 2 are integers. Note that 36 ÷ 2 = 18. Since there are two integers such that when you divide the first integer (36) by the second integer (2) the result is 18, then 18 is a rational number.
A rational number is also a number that can be written as a fraction with integers. For example, 75 is rational since you can express it as 150⁄2, where 150 and 2 are both integers. Also, ½ is a
rational number since it is a fraction composed of integers 1 and 2.
Shown below is the formal definition of a rational number:
A rational number is any number that you can express in the form p⁄q such that p and q are both integers and q is not equal to 0.
Again, don’t worry if the definition sounds confusing because it simply means that a number is a rational number if and only if you can write it as a fraction with integers.
To make it easier for you, here is a list of numbers that can be considered rational numbers:
1. All integers are rational numbers. Example: -1, 0, -1000, 152321, etc. are all rational numbers
2. All fractions (positive or negative) are rational numbers. Example: -5⁄3, -½, 2⁄3 , etc. are rational numbers.
3. All terminating decimals or decimals with an end (positive or negative) are rational numbers: If a decimal number has a finite or countable number of digits, that decimal number is rational.
Example: 0.01, 0.99, -0.23234, -0.421, etc., are rational numbers.
4. All non-terminating (never-ending) repeating decimals (positive or negative) are rational numbers: If a decimal number has an infinite or uncountable number of digits but the digits are being
repeated, then the decimal number is a rational number. Example : 0.9999… , 0.123123123123…, 0.010101…, – 0.11111…, etc. are rational numbers.
Meanwhile, non-terminating non-repeating decimals such as 0.5411346565134…, 0.28992139813…, etc are not rational numbers. Non-terminating non-repeating decimals have an uncountable number of digits
that are not repeated in a certain pattern. You cannot express these non-terminating non-repeating decimals as a fraction with integers.
Note: The “three-dot” symbol that is used on some decimals is called an ellipsis. A decimal with an ellipsis means that there are digits that follow after the last digit of the number. For example,
in 0.9292… the number 2 is not the last digit of the decimal number since there are digits that follow 2.
Irrational Numbers
A number that is not a rational number is an irrational number An irrational number cannot be expressed as a fraction with integers.
Examples of irrational numbers are the non-terminating non-repeating decimals such as 0.321315325453… Using this example, you cannot provide two integers such that when you divide the first integer
by the second integer, the result will be 0.321315325453…
There are a lot of important irrational numbers in mathematics. For instance, the famous pi (π) which is used to calculate the circumference of a circle and the value of which is approximately
3.1416… is an irrational number.
Real Numbers
If you combine the set of rational numbers and the set of irrational numbers, the resulting set is the set of real numbers.
Real numbers are the combination of rational and irrational numbers. -1, 0, 0.12093020…, and -½ are examples of real numbers.
It is clearly seen that the set of the rational numbers and the set of irrational numbers are subsets of the set of real numbers.
The set of rational numbers is composed of the set of integers, fractions, and decimals. On the other hand, the set of positive whole numbers, negative whole numbers, and 0 composed the set of
Therefore, the set of natural numbers, whole numbers, integers, rational, and irrational numbers are all subsets of the set of real numbers.
The Real Number Line
The real number line is a visual representation of the set of real numbers. Every point in the number line corresponds to a real number. Hence, a point in the real number line can be either a
rational or an irrational number.
In the middle of the number line lies the number zero (0). On the left of zero are the negative numbers while all positive numbers are on the right of zero.
The farther you go to the left of the number line, the smaller the value of the number is. As the number line goes to the right, the larger the value of the number is.
For example, – 5 is smaller in value than – 2 since – 5 can be located farther to the left compared to – 2. Meanwhile, 200 is larger in value than 100 since 200 is located farther to the right
compared to 100.
We can also plot fractions and decimals in the real number line. For instance, ½ can be located between 0 and 1, -5⁄2 can be located between -2 and -3, and 4.5 can be located between 4 and 5.
Just like rational numbers, irrational numbers can be located in the real number line. For instance, π whose value is approximately 3.1416 can be located between 3 and 4.
Imaginary Numbers
The first time I heard of imaginary numbers, it felt like hearing about unicorns. You may think that these numbers, just like unicorns, do not exist because they are called “imaginary”.
However, it is important to note that imaginary numbers also “exist” in a mathematical sense and it has practical uses in various fields.
But what exactly is an imaginary number?
An imaginary number is the square root of a negative number. Recall that the square root of a number is the number that when multiplied by itself yields the original number. For example, the square
root of 16 is 4 since 4 x 4 = 16.
However, there’s no real number that gives the square root of a negative number. Suppose that I want to get the square root of -15. -15 has no square root in the set of real numbers since when a real
number is multiplied by itself, the result must always be non-negative.
Mathematical mathematicians used imaginary numbers to express the square root of a negative number. They used i to represent the square root of -1. i is the basic unit of imaginary numbers or the
imaginary unit.
Another interesting fact about imaginary numbers is that they cannot be located in the real number line.
What if we combine a real number and an imaginary number? Well, what you have now is a complex number such as 3 + 5i where 3 is the real number while 5i is the imaginary number.
Next topic: Operations on Integers
Return to the main article: The Ultimate Basic Math Reviewer
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Jewel Kyle Fabula
Jewel Kyle Fabula graduated Cum Laude with a degree of Bachelor of Science in Economics from the University of the Philippines Diliman. He is also a nominee for the 2023 Gerardo Sicat Award for Best
Undergraduate Thesis in Economics. He is currently a freelance content writer with writing experience related to technology, artificial intelligence, ergonomic products, and education. Kyle loves
cats, mathematics, playing video games, and listening to music.
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Common R | Danny James Williams
Common R
Performing operations on vectors
In general, there are three methods that can be used to perform the same operation (such as a mathematical operation) on every element in a vector \(\boldsymbol{x}\). A simple way of doing this is
with a loop, which iterates once for each element in \(\boldsymbol{x}\), and performs the operation one at a time. Vectorisation refers to the process of applying the same operation to every element
in a vector at once, whereas the apply function applies any function across a vector in a single line of code.
Comparison between vectorisation, loops and apply
We can test the efficiency of using vectorisation as opposed to using a loop or an apply function. We will construct three pieces of code to do the same thing, that is, apply the function \(f(x) = \
sin(x)\) to all elements in a vector, which is constructed of the natural numbers up to 100,000. We start by creating the vector:
x = seq(1,100000)
Now we can create three functions. One that uses a for loop, one that uses apply and one that works by vectorising.
loop_function = function(x){
y = numeric(length(x))
for(i in 1:length(x)){
y[i] = sin(x[i])
apply_function = function(x){
y = apply(as.matrix(x), 1, sin)
vector_function = function(x){
y = sin(x)
Now that the functions are constructed, we can use the inbuilt R function system.time to calculate how long it takes each function to run.
## user system elapsed
## 0.058 0.000 0.080
## user system elapsed
## 0.312 0.000 0.312
## user system elapsed
## 0.006 0.000 0.005
Naturally, none of these computations take very long to perform, as the process of taking the sine isn’t very complex. However, you can still see the order of which functions are the fastest. In
general, vectorisation will always be more efficient than loops or apply functions, and loops are faster than using apply.
There are cases where it will not be possible to use vectorisation to carry out a task on an array. In this case, it is necessary to construct a function to pass through apply, or performing
operations within a loop. In general, loops are faster and more flexible - as they allow you to do more in each iteration than a function could. Some situations where you might want to use apply is
to make a simple process neater in the code. If you were doing something relatively straightforward, you will save space and make the code more readable by using apply, as opposed to a loop.
It is common practice to always vectorise your code when you can, as it comes with a significant speed increase, as loops and apply functions are slower than vectorised code.
Other functions
There are different variants of the apply function depending on how your data are constructed, and how you would want your output.
• apply(X, MARGIN, FUN, ...) is the basic apply function. MARGIN refers to which dimension remains constant when performing the function. For example, apply(sum,1,x) will sum across the columns,
and the number of rows will remain constant.
• lapply(X, FUN, ...) is an apply function that returns a list as its output, each element in the list corresponding to applying the given function to each value in X.
• sapply(X, FUN, ...) is a wrapper of lapply that will simplify the output so that it is not in list form.
• mapply(FUN, ...) is a multi-dimensional version of sapply, with mapply it is possible to add more than one input to the function, and it will return a vector of values for each set of inputs.
Map, Reduce and Filter
Map maps a function to a vector. This is similar to lapply. For example:
x = seq(1, 3, by=1)
f = function(a) a+5
M = Map(f,x)
## [[1]]
## [1] 6
## [[2]]
## [1] 7
## [[3]]
## [1] 8
This has added 5 to every element in x, and returned a list of outputs for each element. In fact, Map performs the same operation as mapply does, which we can see in the function itself:
## function (f, ...)
## {
## f <- match.fun(f)
## mapply(FUN = f, ..., SIMPLIFY = FALSE)
## }
## <bytecode: 0x55bfca7455a8>
## <environment: namespace:base>
Reduce performs a given function on pairs of elements in a vector. The procedure is iterated from left to right, and a single value is returned. This can be done from right to left by adding the
argument right=TRUE. As an example, consider division:
f = function(x, y) x/y
x = seq(1, 3, by=1)
Reduce(f, x)
## [1] 0.1666667
Reduce(f, x, right=TRUE)
## [1] 1.5
In the first case, Reduce worked by dividing 1 by 2, then this result by 3. In the second case, this was in reverse, first dividing 3 by 2, then this result by 1.
Filter will ‘filter’ an array into values that satisfy the condition. For example
x = seq(1,5)
condition = function(x) x > 3
## [1] 4 5
Filter is similar to just indexing an array using TRUE/FALSE values, but instead of indexing using an array, it indexes using a function. However, we can inspect the interior of the function
## function (f, x)
## {
## ind <- as.logical(unlist(lapply(x, f)))
## x[which(ind)]
## }
## <bytecode: 0x55bfc77914d8>
## <environment: namespace:base>
So infact, the function for Filter simply uses lapply to get the indices of the TRUE/FALSE values, and indexes the array for input x with a simple subsetting.
Parallel Computing
By using the parallel package, you can make use of all processing cores on your computer. Naturally, if you only have a single core processor, this is irrelevant, but most computers in the modern day
have 2, 4, 8 or more cores. Parallel computing will allow R to run up to this many proccesses at the same time. A lot of important tasks in R can be sped up with parallel computing, for example MCMC.
In MCMC, using \(n\) cores can allow you to also run \(n\) chains at once, with (in theory) no slowdown.
Supercomputers generally have an extremely large number of cores, so being able to run code in parallel is important in computationally expensive programming jobs.
There are some disadvantages to this: namely that splitting a process to four different cores will also require four times as much memory. If your memory isn’t sufficient for the amount of cores that
you are using, this will cause a significant slowdown.
Using mclapply or foreach
There are two main methods to parallelise a set of commands (or a function) in R. The first method is a parallel version of apply, and the second method is a parallel version of mclapply. To
illustrate how these work, consider the example of an \(ARMA(1,1)\) model, which has an equation of the form \[ x_t = \epsilon_t + \alpha\epsilon_{t-1} + \beta x_{t-1} \] \[ \epsilon_t \sim N(0, 1)
\] A function that generates an \(ARMA(1,1)\) process can be written as:
arma11 = function(alpha=0.5, beta=1, initx, N=1000){
x = eps = numeric(N)
x[1] = initx
eps[1] = rnorm(1,0,1)
eps[2] = rnorm(1,0,1)
x[2] = eps[1] + alpha*eps[2] + beta*x[1]
for(i in 3:N){
eps[i] = rnorm(1,0,1)
x[i] = eps[i] + alpha*eps[i-1] + beta*x[i-1]
This will generate a vector of \(x\) values for each timestep from \(t=1,\dots,N\). We can see a plot of this generated time series by running a simulated \(ARMA\) timeseries of length \(N=1000\).
plot(1:1000, arma11(initx=0.5,N=1000),type="l", xlab="Time (t)", ylab=expression(x), main="Time Series Example")
Now that the functions are set up for testing, we can now set up the computer to work in parallel. This involves loading the required packages and detecting the number of cores we have available.
no.cores = detectCores()
## [1] 8
So we have this many cores to work with (on my laptop, there are 8 cores). The no.cores variable will be passed into the parallel computing functions. We can now use mclapply to simulate this \(ARMA
\) model a large amount of times, and calculate the difference from the first value and the last value as a statistic. By putting the arma11 function in a wrapper, we can pass it through to mclapply.
arma_wrapper = function(x) {
y = arma11(alpha=1, beta=1, initx=x, N=1000)
return(head(y,1) - tail(y,1))
xvals = rep(0.5,1000)
MCLoutput = unlist(mclapply(xvals, arma_wrapper, mc.cores = no.cores))
So now, MCLoutput is a vector, of length 1000, that contains the differences between the first and last value in a generated time series from an \(ARMA(1,1)\) model with \(\alpha=1\) and \(\beta=1\).
## [1] -5.373153 -3.740134 -88.112201 53.205845 -4.410669 21.850910
## [1] -0.2121396
Since this process is iterated at every time step, and ran 1000 times on top of that, it will be efficient for testing the efficiency of parallel computing. We can also construct a foreach loop that
will carry out the same task. The foreach function is supplied by the foreach library. It is similar to a for loop but does not depend on each previous iteration of the loop. Instead, foreach runs
the contents of the loop in parallel a specified number of times.
## Loading required package: iterators
FEoutput = foreach(i=1:1000) %dopar% {
y = arma11(initx=0.5,N=1000)
head(y,1) - tail(y,1)
The foreach loop that has been set up performs the same process as the arma_wrapper function earlier at each iteration, it simulates an \(ARMA(1,1)\) process with \(N=1000\) time steps 1000 times.
## [1] -22.303263 -7.588862 20.798046 41.559121 14.506028 29.758766
## [1] -0.8435657
Now that all of the parallel methods are set up, we can time them and compare them to not using parallel at all.
system.time(mclapply(xvals, arma_wrapper, mc.cores = no.cores))
## user system elapsed
## 5.749 0.835 2.056
system.time(foreach(i=1:1000) %dopar% {
y = arma11(initx=0.5,N=1000)
head(y,1) - tail(y,1)
## user system elapsed
## 6.023 0.882 1.847
system.time(for(i in 1:1000){
## user system elapsed
## 4.795 0.007 4.802
This shows that the fastest method is mclapply, which is different from the normal case of apply being slower than a simple loop. Both methods significantly sped up computation time against the
non-parallel version. | {"url":"https://dannyjameswilliams.co.uk/portfolios/sc1/common/","timestamp":"2024-11-05T15:31:33Z","content_type":"text/html","content_length":"33047","record_id":"<urn:uuid:a48c7eba-97e3-451a-b0e2-45c486023b20>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00747.warc.gz"} |
Evolution is a Fact, Just Like Gravity is a Fact! UhOh!
In this week’s New Scientist, there is an article about gravity that deals with a string theorist’s reformulation of gravity as an entropic force. This reformulation describes gravity as an emergent
property of space, time and matter, and NOT as a physical force itself.
Here’s a quote from the actual article:
Of course, Einstein’s geometric description of gravity is beatiful, and in a certain way compelling. Geometry appeals to the visual part of our minds, and is amazingly powerful in summarizing
many aspects of a physical problem. Presumably this explains why we, as a community, have been so reluctant to give up the geometric formulation of gravity as being fundamental. But it is
inevitable we do so. If gravity is emergent, so is space time geometry. Einstein tied these two concepts together, and both have to be given up if we want to understand one or the other at a more
fundamental level.
The results of this paper suggest gravity arises as an entropic force, once space and time themselves have emerged.
I will simply add that my own musings on space-time, and on Einsteinian relativistic presuppositions, tells me that Verlinde is correct, and that he is only one step away (it is a rather giant step,
however) from a great leap forward in our understanding of space, time and matter.
I don’t need to bring out all the implications of this for the Darwinian-ID debate. This theoretical finding throws gravity, as a fundamental force of nature, into question. So, if gravity is now to
be understood as a phenomena, and not a “fact”, then what about Darwinian extravagence?
Jerry, “I was referring to the micro to macro evolution comment” Ok, fair enough. However, it’s important to point out that camanintx made the comparisons based on just genomic differences asking
why one couldn’t have descended from the other. As in, are there that many morphological & other differences that known evolutionary processes couldn’t reasonably account for between what we find
in the fossil record? Quoting you directly : “The incremental and somehow advantageous change from rodent to bat or lizard to bird should be in the fossil record” In this context regarding humans
you’re clearly asking for evidence of fossils which are representative of being intermediary between certain apes and now H. Sapiens. This you admit exists, so question is then are these evidence
of descent w/ modification at the macro level a clearly those exhibited aren't all the same species, so clearly some major changes were going on. “we use a definition of complex novel
capabilities for macro evolution or something similar” Jerry, how would the progression of apes becoming more & more like modern humans over time be for this comparison? Surely, these changes are
novel as they represent the increase in cranium & brain size, changes in morphology which becomes more & more evidently ‘human like’ & less ‘ape like’ would this suffice? If not why not? In terms
of novelty, such increases in the size of a cranium wouldn’t be much different then asking for a group of theropods which had feathers to be related to those which later became the first
primitive birds. I think to some extend the fossil evidence is indicative of what macro evolution aught to entail, never mind such DNA evidence like Human Chromosome 2 fusion.agentorange[
January 27, 2010
08:13 AM
"no evidence it ever happened" I was referring to the micro to macro evolution comment. I just read Dawkins who covers human evolution in fair detail. And lists all the fossils etc. That is not
what I was talking about. I can see where it would be unclear. Just no macro evolution from micro evolution. And if you want to quibble, we use a definition of complex novel capabilities for
macro evolution or something similar. Think eyes, flight, brains etc.jerry[
January 26, 2010
08:09 PM
jerry, #38 Just that there is no evidence it ever happened. ------------------------------------- I suppose a sequence of fossil skulls exhibiting a gradual transition from ape to human over the
last 2.5 million years wouldn't qualify as evidence for you.camanintx[
January 26, 2010
05:04 PM
Jerry, “no evidence it ever happened” You can’t seriously be saying that no such fossils exist which exhibit the intermediate stages that evolution, descent w/ modification, would necessitate,
are you?agentorange[
January 26, 2010
10:44 AM
"The difference between humans and chimps is only about 180 million base pairs. Since these examples illustrate that can and do change over time, maybe you can direct me to a mechanism that
prevents the micro-evolutionary changes illustrated from becoming macro-evolutionary changes over thousands and millions of years." Just that there is no evidence it ever happened. Read Behe's
The Edge of Evolution. Small changes do happen and things can change over a long period of time but so many new capabilities require more than just small changes that getting the new information
for them seems out of range for natural processes. Also gene flow prevents major changes from happening and even with a widely dispersed species such as humans nothing has changed so why when
they were confined to Africa and close to the Chimps were there no gene flow to keep them homogeneous. I can imagine some reasons but to just quote deep time is meaningless. It is more of an
appeal to magic then ID is to an intelligence.jerry[
January 26, 2010
09:08 AM
Davem, #32 “successive incremental changes in groups of organisms from a common ancestor can lead to distinctly separate species.” The incremental and somehow advantageous change from rodent to
bat or lizard to bird should be in the fossil record, but it is lacking. Nor is there the transformation from short to long necked giraffe. Out of all the animals in the world there isn’t a
single example of incremental change in the fossil record. ------------------------------------- I suppose you have never heard of archaeopteryx, the bird that is more like a dinosaur, or
Bohlinia, the giraffe with an intermediate neck length? And maybe you could explain exactly what incremental changes between rodent and bat you think should be present in the fossil record?
January 26, 2010
08:59 AM
jerry, #29 Your examples are micro evolution and don’t make the cut. ------------------------------------- Out of the 3 billion base pairs in our DNA, two human individuals can differ by as much
as a million base pairs. A single individual alone has about 170 mutations in their DNA. With over 6 billion individuals currently living, that is a lot of random mutation for natural selection
to work from. The difference between humans and chimps is only about 180 million base pairs. Since these examples illustrate that can and do change over time, maybe you can direct me to a
mechanism that prevents the micro-evolutionary changes illustrated from becoming macro-evolutionary changes over thousands and millions of years.camanintx[
January 26, 2010
08:24 AM
"It surprises me that you “believe there is a site someplace on the internet that keeps track of them" Here is a site that seems to keep track of them. I actually thought there was a lot more as
I was reading about a guy who was in charge of this but I overestimated the number. http://genomesonline.org/index.htm Here is another site that says there 180 different organisms http://
January 25, 2010
02:47 PM
caminintx wrote:
Researchers have observed organisms gaining new functions through random mutation and natural selection.
I presume you are referring to loss of information, or minimal incremental change, both already well covered by Behe. In any event, you are definitely talking about microevolution, which is my
point precisely -- we have no basis to extrapolate grand Evolution from the trivial and obvious changes seen in microevolution.
They have observed how successive incremental changes in groups of organisms from a common ancestor can lead to distinctly separate species.
Really? You are again talking about microevolutionary changes and using a very minimal "species" definition. I would be very interested in any examples of "successive incremental changes" that
produced any macroevolutionary change. Fact: there aren't any you can point to.
Extrapolating a theory of common descent from these observed facts is no harder than deriving Galileo’s heliocentric theory from Newton’s laws of gravity.
You seem confused about the analogy. How does gravity extrapolate to heliocentrism? In any event, back to the topic, the question is precisely whether the extrapolation of a grand evolutionary
narrative from the paucity of observed facts is warranted. Simply stating that one can extrapolate isn't helpful -- that is precisely the point at issue. We can just as easily assert, as I will,
that there is no good reason to think that we can extrapolate broader macroevolutionary changes from the observed microevolutionary changes. So far, it appears there is no evidence to support
such an extrapolation, only that the theory needs it.Eric Anderson[
January 24, 2010
11:26 PM
It seems all their data for complexity arising is in computer programs and not actual genomes. You would think that genomes is where the action should be.
You may tell this the sciensists at the Biologic Institute and the Evolutionary Informatics Lab who are focussing on algorithms too.osteonectin[
January 24, 2010
09:17 PM
24 camanintx "successive incremental changes in groups of organisms from a common ancestor can lead to distinctly separate species." The incremental and somehow advantageous change from rodent to
bat or lizard to bird should be in the fossil record, but it is lacking. Nor is there the transformation from short to long necked giraffe. Out of all the animals in the world there isn't a
single example of incremental change in the fossil record.Davem[
January 24, 2010
09:12 PM
I think several thousand have been sequenced. I believe there is a site someplace on the internet that keeps track of them. My guess is the software to analyze them and to compare them is
what is at issue. Also I am sure there are many more partial genomes that have been sequenced. Nobody has reported anything earth shattering yet for all these sequenced genomes both whole and
It surprises me that you "believe there is a site someplace on the internet that keeps track of them". Actually, these sequenced genomes should be at the heart of ID research and should be used
as the main resource for anybody really wanting to do ID research. Unfortunately, neither the Biologic Institute nor the Evolutionary Informatics Lab went in this direction (to my best knowledge
they didn't publish anything in this field). Currently, all sequencing data are assembled by programs that presume evolution as the source of biological information. And these programs run
reliably as has been proven by molecular biological techniques e.g. FISH staining of chromosomes [not to be mistaken as your inner fish ;)], genomic southerns, radiation hybrid clones etc. again
and again. It would be interesting to see what design theory could contribute to this field because there are some fields where better algorithms would be helpful. If ID would produce anything
that could be proven in the wet lab reality the community would quickly adopt such techniques. Until then you may be more reluctant to convict 21st century biology.osteonectin[
January 24, 2010
09:11 PM
camanintx[21]: You're indulging in the very kind of rationalization I decry. You're giving a "know-it-all" answer. You think you know something and you know nothing. Read Verlinde's article and
try to understand it. Here's what he says in the "Conclusion and Discussion" section: (This is the very first paragraph.)
Gravity has given many hints of being an emergent phenomenon, yet up to this day it i still seen as a fundamental force. The similarities with other known emergent phenomena, such as
thermodynamics and hydrodynamics, have been mostly regarded as just suggestive analogies. It is time we not only notice the analogy, and talk about the similarity, but finally do away with
gravity as a fundamental force."
Do you get it now? Gravity is, as I've stated above, an "epiphenomenon", not a real, fundamental force. Rather, it is a byproduct of a different kind of physical architecture; that is, the net
result of "entropic forces" (which are microscopic and involve "degress of freedom", none of which involves "mass", except for its effect on these degrees of freedom). This means that the whole
way of thinking about gravity must change. Yes, apples still fall in "perpendicular lines" to the earth, as Newton noticed, but the "force" he associated with those "perpendicular lines" really
doesn't exist---just as Darwinian evolution doesn't exist, although the fossil record shows a progressives "evolution" of forms. But happy rationalizing! I'm sure you're not going to want to
change the way you think about gravity or Darwin. And, in this, truth suffers.PaV[
January 24, 2010
09:00 PM
"I’m sure they can find these themselves." Your examples are micro evolution and don't make the cut. I do not know how many times the nylon example has been brought up but it is irrelevant. ID
doesn't question changes like these. That you brought them up here, means you don't have anything. So I take back my suggestion and that you should not contact Dawkins or Coyne. You are no better
off than they are.jerry[
January 24, 2010
08:58 PM
jerry, #27 “Researchers have observed organisms gaining new functions through random mutation and natural selection. They have observed how successive incremental changes in groups of organisms
from a common ancestor can lead to distinctly separate species.” Then maybe you should write a book on it. I am over half way through Dawkins’ new book and so far he hasn’t reported any of this.
You should email him so you can help him with his next book. Also email Jerry Coyne and let him know too. ---------------------------------- I'm sure they can find these themselves. Appl.
Environ. Microbiol., May 1995, 2020-2022, Vol 61, No. 5 Copyright © 1995, American Society for Microbiology Emergence of nylon oligomer degradation enzymes in Pseudomonas aeruginosa PAO through
experimental evolution ID Prijambada, S Negoro, T Yomo and I Urabe and Nature 230, 289 - 292 (02 April 1971); doi:10.1038/230289a0 Experimentally Created Incipient Species of Drosophila
Theodosius Dobzhansky & Olga Pavlovskycamanintx[
January 24, 2010
06:04 PM
"Researchers have observed organisms gaining new functions through random mutation and natural selection. They have observed how successive incremental changes in groups of organisms from a
common ancestor can lead to distinctly separate species." Then maybe you should write a book on it. I am over half way through Dawkins' new book and so far he hasn't reported any of this. You
should email him so you can help him with his next book. Also email Jerry Coyne and let him know too.jerry[
January 24, 2010
05:32 PM
"Since only a few biological genomes have been completely sequenced, how long do you suggest we wait for experiments based on them to be developed, executed and analyzed" I think several thousand
have been sequenced. I believe there is a site someplace on the internet that keeps track of them. My guess is the software to analyze them and to compare them is what is at issue. Also I am sure
there are many more partial genomes that have been sequenced. Nobody has reported anything earth shattering yet for all these sequenced genomes both whole and partial.jerry[
January 24, 2010
05:30 PM
jerry, #23 It seems all their data for complexity arising is in computer programs and not actual genomes. You would think that genomes is where the action should be.
------------------------------------ Since only a few biological genomes have been completely sequenced, how long do you suggest we wait for experiments based on them to be developed, executed
and analyzed?camanintx[
January 24, 2010
03:39 PM
Eric Anderson, 22 Sorry, but you must be *way* more specific than that. I am aware of many different ways the word “evolution” is used in the literature, ranging from the obvious and
well-observed to the outrageous and wildly-speculative. For example, if all you mean is that nature isn’t forever static, then big deal — and we don’t need Charles’ theory to help us there. If,
on the other hand, you mean something much more grand (and controversial), such as abiotic origins, descent of man, universal common descent, etc., then you have no “fact” to point to — it is
still theory. ----------------------------------- It's really not that complicated. The Theory of Evolution is based upon mechanisms derived directly from observed facts. Researchers have
observed organisms gaining new functions through random mutation and natural selection. They have observed how successive incremental changes in groups of organisms from a common ancestor can
lead to distinctly separate species. Extrapolating a theory of common descent from these observed facts is no harder than deriving Galileo's heliocentric theory from Newton's laws of
January 24, 2010
03:34 PM
I went to the Adami article referenced by caminintx and then to the articles citing it and it a veritable gold mine of stuff on complexity and information with lots of full text articles. Here is
the link I accessed http://scholar.google.com/scholar?q=link:http%3A%2F%2Fwww.pnas.org%2Fcgi%2Fcontent%2Fabstract%2F97%2F9%2F4463 It seems all their data for complexity arising is in computer
programs and not actual genomes. You would think that genomes is where the action should be.jerry[
January 24, 2010
11:21 AM
caminintx wrote: "There is the fact of evolution, that populations of organisms change over time . . ." Sorry, but you must be *way* more specific than that. I am aware of many different ways the
word "evolution" is used in the literature, ranging from the obvious and well-observed to the outrageous and wildly-speculative. For example, if all you mean is that nature isn't forever static,
then big deal -- and we don't need Charles' theory to help us there. If, on the other hand, you mean something much more grand (and controversial), such as abiotic origins, descent of man,
universal common descent, etc., then you have no "fact" to point to -- it is still theory. There is simply no comparison between evolution and gravity in terms of their factual content,
specificity, or predictive capabilities. Gravity is observed literally every moment of every day; we can measure very precisely how planes and rockets will react at any given distance from the
earth; we can send men to the moon and ships to the outer planets. Sure, there are some very interesting nuances about exactly how gravity works, and we probably have much to learn there. But for
someone to suggest that evolution (whatever that slippery term means) is as well an established fact as gravity is more than ludicrous, it is laughable.Eric Anderson[
January 24, 2010
10:44 AM
PaV, #20 When an argument is presented as being the strongest of arguments, and then this argument is shown to be suspect, to simply go on as if it’s no big deal reeks of rationalization. When
that happens, truth suffers. ------------------------------------- Newton's theory of gravity became suspect in 1859 when Mercury's orbital precession was found to differ from what the theory
predicted. Despite the advent of Einstein's Theory of Relativity, Newtonian physics is still taught today because it is foundational. Has truth suffered because of this?camanintx[
January 24, 2010
08:41 AM
Heinrich: I speak Italian. Yes, the effects of gravity are real; however, the causal mechanism for gravity, per Verlinde, is illosory if we believe either Newton or Einstein. Is Darwin so
sacrosanct? When an argument is presented as being the strongest of arguments, and then this argument is shown to be suspect, to simply go on as if it's no big deal reeks of rationalization. When
that happens, truth suffers. bornagain: Interesting quotes. I'll look at your link. I'm very impressed with Penrose.PaV[
January 23, 2010
08:41 AM
REC -- but common descent is well observed. Intra-species, maybe, (i.e. dogs) but are you saying a prokaryote has been seen evolving into a eukaryote?tribune7[
January 23, 2010
06:16 AM
Translation: gravity is illusory.
E pur si muove.Heinrich[
January 22, 2010
03:32 PM
uoflcard, #15 Of course it does not reject Darwinism because variation and selection do happen. See malaria resistance, for example. So Darwinism is part of the extended synthesis because it does
do things. They just need to conjure up a natural way to produce complex, specified information. ------------------------------------------------ Many have already done that. Evolution of
Biological Information Evolution of biological complexity The evolutionary origin of complex featurescamanintx[
January 22, 2010
02:22 PM
PaV, Of interest,,, The Physics of the Small and Large: What is the Bridge Between Them? Roger Penrose Excerpt: "The time-asymmetry is fundamentally connected to with the Second Law of
Thermodynamics: indeed, the extraordinarily special nature (to a greater precision than about 1 in 10^10^123, in terms of phase-space volume) can be identified as the "source" of the Second Law
(Entropy)." http://www.pul.it/irafs/CD%20IRAFS%2702/texts/Penrose.pdf That statement was from a paper in which Penrose was trying to find a bridge between quantum mechanics and gravity,,,,and
tends to agree with Verlinde's statement:
The results of this paper suggest gravity arises as an entropic force, once space and time themselves have emerged.
,,, After a bit of reflection,, I think there is much promise in his "holographic" model for more precisely elucidating the required "non-material" explanation for gravity, or at least for
elucidating how space-time curvature precedes as well as is primary and dominant of "material" gravity. i.e. His model seems to be "primed" to account for "dark matter",,, and indeed his model
may also be found to explain dark energy as well if he can find a congruent entropic measurement for Dark Energy, which at first glance should be so since Dark Energy is "expanding" space-time
January 22, 2010
08:48 AM
#4 camanintx
It will be titled, Evolution: The Extended Synthesis. “The word ‘extended’ is important because it implies quite clearly that there is no rejection of the previous synthesis,” Pigliucci
says. “There is no rejection of the Modern Synthesis. There is no rejection of Darwinism. It’s an extension of it—we think a significant extension in a lot of different directions which
neither Darwin nor the Modern Synthesis could have possibly thought of.”
This doesn’t exactly support your belief that that population genetics (= to Modern Synthesis, more or less) is “a complete waste of time”.
Of course it does not reject Darwinism because variation and selection do happen. See malaria resistance, for example. So Darwinism is part of the extended synthesis because it does do things.
They just need to conjure up a natural way to produce complex, specified information. So Darwinism and the Modern Synthesis should make up around 0.001% of the Extended Synthesis - at least as
far as the production of new information is concerned (it does a great job preserving what is already there). Hence the phrase "significant extension" - if Darwinism were the main driving force
behind biological novelty, an extension wouldn't need to be labeled "significant". So as for the origin of life and species, yes I would comfortably say that population genetics is a waste of
January 22, 2010
06:50 AM
Eric Anderson: I share your same concerns. REC: bornagain is not a troll. And his posts don't usually have this same kind of content. He's a valuable contributor here. bornagain: The ideas you're
expressing are similar to mine. I have a definite notion of how some of physics can be reinterpreted and have started to pursue it a bit. It could prove to be very interesting. And....it would
have theological implications, though that is not the starting point of my reflections. Retroman: Two points: (1) The argument is routinely made that evolution is a "fact" just like gravity is a
"fact". This argument is meant to say that evolution is, in fact, unassailable. (2) The counterargument has always been that there are several ways to consider evolution. For example, there is
the fossil record. There are "breeds" that can be formed, thus indicating plasticity of phenotype. And then there is the explanation given for all these phenomena, which is itself called
"evolution". And, so, while obviously observing the phenomena, we can still reject the explanatory cause given. Our critics arrogantly reject this counterargument. Verlinde's treatment of gravity
shows just how arrogant---and unscientific---our critics have been. AND, further, they have sought refuge on this island of surety, and that island is being shaken to its roots. Verlinde's
interpretation of gravity is not a force at a distance like Newton, nor is it, exactly, a curvature of space like Einstein, but rather has to do with the characteristics of space itself, and,
specifically, its many degrees of freedom coupled to its ultimate discreetness. So, the explanatory core of gravity has changed---radically; something our critics would have laughed at us about
should we have proposed it. It's now our turn to laugh at their arrogance. ALL theories are ultimately hypotheses that have, at the moment, grounds for acceptance. But that can always change. And
science has a history of theories coming and going. But NOT Darwinism. So what if his theories are 160 years old. So what if the "co-founder" of natural selection ended up in disagreement with
Darwin. So what if the Modern Synthesis is completely incapable of explaining what we find in nature. No matter---the theory lives on. A little humility, I think, is now in order.PaV[
January 22, 2010
06:37 AM
PaV, #6 And, please, tell me how you feel about the argument that has been made by so many for so long; namely, evolution, like gravity, is a fact. Well, the chair has been pulled out of that
one. It really is quite an embarassment for the Darwinist side–or, at least, it should be. --------------------------------------- There is the fact of gravity, that things fall when I drop them,
and the Theory of Gravity, that things fall because masses attract each other. There is the fact of evolution, that populations of organisms change over time, and the Theory of Evolution, that
these changes are explained by random mutations and natural selection. Darwin's Theory of Evolution was certainly not the be-all and end-all of evolutionary theory just as Newton's Theory of
Gravity was not the last word in physics. Modern Synthesis and Extended Synthesis no more discard RM & NS as a process of biological evolution than Relativity discarded Newton's Laws of motion,
they merely complete them.camanintx[
January 22, 2010
06:29 AM
1 2 Next
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Be conscious of your movement vector
We are multi-dimensional beings. I speak of dimensions comprising our physical and psychological attributes, hobbies, interests, habits, humor, taste for 20th-century classicism, etc. The individual
dimensions number too many to list.
Now imagine yourself as a point on this highly-dimensional plane. The points closest to you are probably your friends. I say that backwards. Rather, your friends - real friends, that is, are probably
among the points closest to you. However, the higher the number of dimensions you project on the plane, the less neighbors you will find around you. It can get lonely on that plane.
Okay, let me back up. For simplicity sake, suppose we are nothing but primitive creatures of two dimensions. You can now visualize this plane and even sketch it on paper. You can scatter a bunch of
points on the plane to represent various people you know or don’t know. This is entirely hypothetical.
Let’s think of the unit density as the average number of points per unit hyperplane. In 2-dimensions, this unit hyperplane is a simple square of arbitrary unit area. In 3-dimensions this becomes a
cube. And so on. The greater the density, the more neighbors you will find, on average, and the more trusted friends you probably have.
Suppose this 2D world now becomes self-aware. The inhabitants realize there’s more to them than representable in two dimensions. Suddenly they conceptualize a third dimension. Bear in mind, the
number of inhabitants remains constant, but we now project them onto three dimensions. Guess what? The number of neighbors in your unit hyperplane (a cube) is now notably less. That is, assuming you
populated your world not with your clones but with inhabitants of sufficient variety and uniformity. Convince yourself of this fact.
It gets worse. With each increasing dimension, the unit density falls exponentially.
But all mathematical relation aside, the greater the number of dimensions, the less beings in common you will find among your network. You can probably induce this observation to what you know about
human beings, compared to say, animals. The loneliness, the solitude, are especially common among those singular beings of particular taste and inclination.
Bear in mind, this dimensional projection of yourself is nothing more than that, a projection. If you have certain friends with whom you share an interest in vinyl records, no multi-dimensional
projection will alter that. You will still align among that one dimension. And yet, the higher the dimensionality we imagine ourselves to span, the more complex of an identity we project among
ourselves, and the greater difficulty we’ll face in accepting others into our trusted domain. You may align among one dimension, but your singularity suddenly skyrockets in a world that evolves from
three to ten or a hundred total dimensions.
If you find yourself victim to this curse of dimensionality, find a way to collapse your world to a smaller number of dimensions. It may not alleviate your existentialism struggle, but it will
certainly ease the task of reestablishing a trusted circle of contacts.
To further complicate matters, we occupy a world of continuous movement. As points on a plane, we generally don’t remain static. At least in some dimension we find ourselves in gradual movement
across time. And this likely is the case across a number of dimensions.
With that noted, our identity is better defined not strictly as a point on a plane, but as a vector; a vector of movement; a vector that illustrates our direction within some discrete time interval.
The reliability of our trusted group of friends is now better represented not only by the proximity to each other on the plane, but by the similarity between the vectors.
Let’s take one dimension of commonality, the vinyl records. You may find yourself in relative proximity on that axis with certain friends. But your movement among the axis over time may differ. Their
interest may gradually increase, while yours slowly diminishes. With time, you lose commonality along that dimension.
With a higher number of dimensions, the similarity between the movement vectors of the world inhabitants also falls. At one instant, we may coincide across many dimensions even. And yet our vectors
may entirely differ. We may be arriving from different points, roughly coincide for a discreet time interval, and in but a brief moment set course for complete divergence.
Be conscious of your movement vector. Communicate about it. It is fundamental to reaching mutual understanding. It is representative of the likelihood that your network and your friends survive the
test of time.
Questions, comments? Connect. | {"url":"https://vitalyparnas.com/posts/movement-vector/","timestamp":"2024-11-10T15:42:30Z","content_type":"text/html","content_length":"6546","record_id":"<urn:uuid:90b909ba-63c1-4181-a64d-9848f05ec356>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00870.warc.gz"} |
Symmetric and asymmetric encryption - Moment For Technology
Encryption is widely used in programming, especially in various network protocols. Symmetric/asymmetric encryption is often mentioned as two encryption methods.
Symmetric encryption
The vast majority of encryption that we encounter at ordinary times is symmetrical encryption, for example: fingerprint unlock, PIN code lock, safe combination lock, account password is used
symmetrical encryption.
Symmetric encryption: encryption and decryption using the same password or the same logical encryption method.
This is also known as a symmetric key, but symmetry and asymmetry refer to whether the encryption and decryption keys are the same.
When I was in college, I made a command line version of the library management system as a C language course. When you log in to the system, you need to input your account password. Of course,
verifying the password entered by the user is a kind of symmetric encryption. The password must be the same as the account password you set before.
I chose to store the password in a local file, but if the file is stolen and the password itself is not encrypted, the password is compromised. So how do you encrypt passwords stored in files? The
way I adopted at that time (which can also be understood as an encryption algorithm) was to take the code value of each character of the account password set by the user and add a fixed value, such
as 6, and then store the calculated code value string when storing. Use JS to simulate:
// Process of encrypting user passwords
const sourcePassword = 'abcdef'; // User account password
// Encrypted password of the account stored in the local file
const encryptedPassword = [...sourcePassword].map((char) = > char.codePointAt(0) + 6).join();
console.log('The encrypted account password is:${encryptedPassword}`) // => the encrypted password is 103,104,105,106,107,108
// Decryption process
const decryptedPassword = encryptedPassword.split(', ').map((codePoint) = > String.fromCodePoint(codePoint - 6)).join(' ');
console.log('Decrypted account password is:${decryptedPassword}`); // => The password of the decrypted account is abcdef
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The above procedure for encrypting user account passwords does not obviously involve the password used for encryption, but the encryption and decryption use the same logic: take the ASCII value of
the character plus 6, in fact, this is symmetric encryption. In fact, it can also be understood that the encryption password is the encryption algorithm itself, and the encryption algorithm can
decode the account password.
Some software supports data encryption, such as RAR, which requires you to input the password when encrypting data, and then requires us to input the set encryption password when decrypting data.
Actually going back to the very simple encryption algorithm I mentioned above, the system can be designed so that the user can set an encrypted password to encrypt the account password. This is
reduced to a single number, and instead of adding a fixed 6 to each user’s password, the system will add the number set by the user. Now your encryption logic can be disclosed as rar compression
algorithm, even if the attacked know the encryption algorithm and the encrypted account password, if they do not know the encryption password, they still cannot get the user account password. Here
the encryption password and decryption password are the same password, corresponding to the above system is 6, so also symmetric encryption.
Use NodeJS for symmetric encryption
Nodejs’s crypto module is a dedicated module for all kinds of encryption, can be used to extract (hash), salt digest (HMAC), symmetric encryption, asymmetric encryption, etc. Symmetric encryption
with Crypto is simple, with the Crypto module providing Cipher classes for encrypting data and Decipher for decryption.
Common symmetric encryption algorithms include DES, 3DES, AES, Blowfish, IDEA, RC5, and RC6. The AES algorithm is used in this section.
const crypto = require('crypto');
/** * symmetric encryption String * @param {String} password Secret key used for symmetric encryption * @param {String} String Encrypted data * @return {String} encryptedString The encrypted character string */
const encrypt = (password, string) = > {
// The symmetric encryption algorithm is AES-192-Cbc
const algorithm = 'aes-192-cbc';
// Generate symmetric encryption key, salt is used to generate the key, 24 specifies that the key length is 24 bits
const key = crypto.scryptSync(password, 'salt'.24);
console.log('key:', key); // => key: <Buffer f0 ca 6c ac 39 3c b3 f9 77 13 0d d9 bc cb dd 9d 86 f7 96 e0 75 53 7f 8a>
console.log('Secret key length:${key.length}`); // => Length of key: 24
// Initialize the vector
const iv = Buffer.alloc(16.0);
// Get the Cipher Cipher class
const cipher = crypto.createCipheriv(algorithm, key, iv);
// UTF-8 specifies the character encoding of the data to be encrypted, and hex specifies the character encoding of the output
let encryptedString = cipher.update(string, 'utf8'.'hex');
encryptedString += cipher.final('hex');
return encryptedString;
const PASSWORD = 'lyreal666';
const encryptedString = encrypt(PASSWORD, 'Talent doesn't work hard enough.');
console.log('The encrypted data is:${encryptedString}`); / / = > the encrypted data is: 1546756 bb4e530fc1fbae7fd2cf9aeac0368631b54581a39e5c53ee3172638de
/** * decryptedString * @param {String} password encryption password * @param {String} encryptedString encryptedString * @return {String} decryptedString Decrypted string */
const decrypt = (password, encryptedString) = > {
const algorithm = 'aes-192-cbc';
// Use the same algorithm to generate the same secret key
const key = crypto.scryptSync(password, 'salt'.24);
const iv = Buffer.alloc(16.0);
// Generate Decipher decryption class
const decipher = crypto.createDecipheriv(algorithm, key, iv);
let decryptedString = decipher.update(encryptedString, 'hex'.'utf8');
decryptedString += decipher.final('utf8');
return decryptedString;
const decryptedString = decrypt(PASSWORD, encryptedString);
console.log('Decrypted data:${decryptedString}`); // => Decrypted data: talent is not hard enough to gather together
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Asymmetric encryption
Asymmetric encryption uses a pair of secret keys, called a public key and a private key, also known as an asymmetric secret key. Asymmetric keys can be used for both encryption and authentication.
Let’s talk about encryption first.
One password is enough for encryption, why two passwords for asymmetric encryption?
Black question mark. JPG
I’m sure there are people out there who have had that thought. In fact, symmetric encryption as long as the length of the encrypted password is long enough, the encrypted data is generally secure
without access to the password itself. But there is a problem in the practical application of network data such as encryption, because use the same secret key encryption and decryption, so the server
and the client is necessarily to the exchange of the secret key, and it is because of the asymmetric secret key with a secret key exchange this process may be an intermediary to steal the secret key,
once the symmetrical secret key stolen, and by analyzing the encryption algorithm, The transmitted data is then transparent to the middleman. So the fatal drawback of symmetric encryption is that it
cannot guarantee the security of the secret key.
So does asymmetric encryption guarantee the security of a secret key? Yes, the secret key can be made public, and the secret key that is made public is called a public key. Asymmetrically encrypted
secret keys are calculated by encryption algorithms and come in pairs. The public key is called a public key, and the private key that cannot be made public is called a private key.
When using github and other Git repository hosting platforms, we usually configure SSH public key, generate a pair of secret keys we can use the following command:
The above uses the ssh-keygen program to specify the encryption algorithm as RSA, and generates a pair of key, key.pub in the current directory. Pub = public key = public key = public key Take a look
at the generated content:
The key.pub is a public key, and public keys are designed to be public at will. By configuring this public key to the hosting platform, we can eliminate the need to enter a password every time we
communicate with Github.
The key to the security of asymmetric encryption is to use one secret key in a private key pair, and the encrypted data can only be decrypted through the other secret key. This means that data is
encrypted using the public key of a pair of secret keys and can only be decrypted using the other private key. Or conversely, data encrypted using the private key in a pair of secret keys can only be
decrypted using the other public key. Thus, from the point of view of encryption, public key and private key are the same function, can be used for encryption or decryption, but when we use
asymmetric secret key for encryption data is often used for encryption with public key.
In HTTPS encryption, symmetric encryption is used to encrypt the transmitted data itself, and asymmetric encryption is used to encrypt symmetric secret keys. The whole process is as follows: The
server creates a pair of asymmetric secret keys and sends the public key to the client. The client also determines the symmetric encryption algorithm and the symmetric secret key used for data
transmission, and then encrypts the symmetric secret key using the public key given by the server. After receiving the symmetric encryption algorithm and key from the client, the server and client
use the symmetric encryption algorithm and key for data transmission. Without the private key stored on the server side, you cannot decipher the symmetric secret key encrypted with the public key,
even if the contents of the public key on the server side are known to the middleman. A public key is a public key that can be accessed at will, and decryption requires a private key. Asymmetric
encryption or public key encryption can ensure encryption security because the private key is not public, attackers can not crack without the private key.
One might wonder: why use asymmetric encryption to encrypt symmetric keys? That’s because the exchange of symmetric keys can be stolen by a third party, and if symmetric keys are stolen then
symmetric encryption makes no sense. And why not just use asymmetric encryption to encrypt the transmission instead of just the symmetric key? Asymmetric encryption isn’t symmetric encryption more
secure? This is related to the characteristics of symmetric encryption and asymmetric encryption.
Contrast asymmetric encryption with symmetric encryption
1. Symmetric encryption is one secret key, asymmetric encryption is a pair of two secret keys
2. Asymmetric encryption is more secure than symmetric encryption because there is no leakage of the secret key and the public key is known
3. Because asymmetric encryption is computationally complex, symmetric encryption and decryption are generally much faster than asymmetric encryption
4. Asymmetric keys can also be used for authentication
Asymmetric encryption is not used for data transmission in HTTPS because the data itself may be large. In this case, asymmetric encryption takes more time. Therefore, asymmetric encryption is not
used for data transmission.
Use NodeJS to demonstrate asymmetric encryption
Common asymmetric encryption has RSA, ECC (elliptic curve encryption algorithm), Diffie-Hellman, El Gamal, DSA (digital signature), here demonstrates RSA encryption.
const crypto = require('crypto');
// Key phrase
const passphrase = 'lyreal666';
// rsa Indicates that the asymmetric key algorithm is RSA
const { publicKey, privateKey } = crypto.generateKeyPairSync('rsa', {
modulusLength: 4096.// Specify the length of the key
publicKeyEncoding: {
type: 'spki'.// Public key encoding format
format: 'pem',},privateKeyEncoding: {
type: 'pkcs8'.// Private key encoding
format: 'pem'.cipher: 'aes-256-cbc',
/** * encrypting with publicKey * @param {String} publicKey the secret key used for symmetric encryption * @param {String} String the encrypted data * @return {String} encryptedString The encrypted character string */
const encrypt = (publicKey, string) = > {
// Use public key encryption
return crypto.publicEncrypt({ key: publicKey, passphrase } , Buffer.from(string)).toString('hex');
/** * encryptedString with privateKey * @param {String} privateKey privateKey * @param {String} encryptedString DecryptedString indicates the decryptedString */
const decrypt = (privateKey, encryptedString) = > {
return crypto.privateDecrypt({ key: privateKey, passphrase } , Buffer.from(encryptedString, 'hex'));
const string = 'Say don't cry, don't love me, cry (' ⌒´) Blue';
const encryptedString = encrypt(publicKey, string);
console.log('Result of public key encryption:${encryptedString}`); // => Result of public key encryption: caf7535C46146f5...
const decryptedString = decrypt(privateKey, encryptedString);
console.log('Decrypted private key result:${decryptedString}`); // => Private key decrypt result: say, don't cry, don't love me, cry (' ⌒´) Blue
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Asymmetric key authentication
Asymmetric encryption is sometimes called public key encryption, and asymmetric secret key authentication is also called private key authentication. When we say we use asymmetric keys to authenticate
data, what we mean is to verify that data has not been tampered with. Asymmetric keys are widely used for authentication in addition to encrypting data, such as mobile apK signatures and certificates
in HTTPS.
The principle is simple: for example, if I want to verify that an APK code has been modified, I first prepare a pair of asymmetric secret keys, usually from an authority. The official apK package
includes not only the application code, but also a signature, which is simply interpreted as the encrypted data using the hash value of the application code using the private key. When the APK is
installed, the Android system extracts the signature in the APK, decrypts the signature using the public key to obtain the hash of the original application code, and compares the hash with the hash
of the original application code. If the content is the same, the APK has not been tampered with. If apK’s application code has been modified by a third party, the hash decrypted from the signature
must be different from the hash of the application code. So you can make sure that the application code is not tampered with, which is authentication.
The key to authentication is the presence of the signature, which must be guaranteed to hash the original APK application code. How to ensure that signatures are not tampered with is beyond the scope
of this article.
Some people may look at the apK authentication process will have such a question: using the private key to encrypt the content can achieve the purpose of authentication, that can use the public key
encryption to authenticate?
The answer is definitely no. If you use a public key to encrypt your content, the middleman will have to tamper with your content. Forging a signature is as simple as using the public key to hash the
forged content. So another key to using asymmetric keys for authentication is that the private key is not public, so middlemen can’t get access to the private key and therefore can’t forge the
A few questions
Is hash encryption?
I don’t think so. Hash is irreversible and the encryption should be able to be restored based on the encrypted data.
Is Base 64 encryption?
Symmetric encryption, symmetric key is the base 64 character code table.
Is asymmetric encryption absolutely safe?
No encryption is absolutely secure, and asymmetric encryption has the problem of public keys being tampered with when they are exchanged.
Because I have not studied cryptography, the above content is written by me through summarizing the knowledge I have learned in the past, it is inevitable that there will be some bias, even said with
a very obvious personal point of view. Readers are welcome to point out errors and discussions in the comments section.
Thank you for reading. If it is helpful to you, please pay attention and click a “like” to support it.
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The Weekly Challenge - Perl & Raku
| Day 6 | Day 7 | Day 8 |
The gift is presented by Simon Green. Today he is talking about his solution to “The Weekly Challenge - 157”. This is re-produced for Advent Calendar 2022 from the original post by him.
Weekly Challenge 157
I’m back after the three week hiatus!
[My solution]
TASK #1 › Pythagorean Means
You are given a set of integers.
Write a script to compute all three Pythagorean Means i.e Arithmetic Mean, Geometric Mean and Harmonic Mean of the given set of integers.
My solution
This is a relatively straight forward task. The Wikipedia page provides the necessary formulas and we can use the sum, reduce and lambda functions to calculate the required figures. Finally we use
the round function to print the results to one decimal place.
The Perl code is similar to the Python version, except we use the product() method in List::Utils. Python's math.product is only available for Python 3.8+
One thing to note is if the sum of integers is 0, a division by zero error might occur when calculating the harmonic mean. I’m not catching this and let the error pass to the user.
$ ./ch-1.py 1 3 5 6 9
AM = 4.8, GM = 3.8, HM = 2.8
$ ./ch-1.py 2 4 6 8 10
AM = 6.0, GM = 5.2, HM = 4.4
$ ./ch-1.py 1 2 3 4 5
AM = 3.0, GM = 2.6, HM = 2.2
TASK #2 › Brazilian Number
You are given a number $n > 3.
Write a script to find out if the given number is a Brazilian Number. A positive integer number N has at least one natural number B where 1 < B < N-1 where the representation of N in base B has [the]
same digits.
My solution
This was an interesting task. We can’t convert numbers into a different base simply by using letters and numbers. For example 1282 is 22 in base 640, but how would you express 639 in the same base?
For this task I considered a value having the same digits even if the number itself has different digits. For example, 925 is express as (25)(25) in base 36.
The guts of this task is to convert the number n in base 10 to a specified base, which I have called b. The function is called same_digits. I take the modulus (remainder) of n and b and make this
value d. I then do an integer division and repeat this process comparing that n % b is d. I return False if any digit isn’t the same, or True if they all are.
Then it’s just a matter of having a loop from 2 to n - 2, and calling the same_digits function for each base. If there is a match, I print 1 (and the base), or 0 otherwise.
The Perl code is a transliteration of the Python code.
$ ./ch-2.py 6
$ ./ch-2.py 7
1 (base 2)
$ ./ch-2.py 8
1 (base 3)
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